CATEGORICITY OF AN ABSTRACT

ELEMENTARY CLASS IN

TWO SUCCESSIVE CARDINALS

SH576

Saharon Shelah

Institute of Mathematics The Hebrew University Jerusalem, Israel

Rutgers University Department of Mathematics New Brunswick, NJ USA

Abstract. We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models, or existence of large cardinals). We prove (assuming a weak version of GCH around λ) that if 𝔎 is categorical in λ, λ^{+}, LS(𝔎) ≤ λ and has intermediate number of models in λ^{++} \underline{then} 𝔎 has a model in λ^{+++}.

1991 Mathematics Subject Classification. 1988 math subject classification 03C45,03C75, 03C95. Key words and phrases. model theory, abstract elementary classes, classification theory, cate-

goricity, nonstructure theory.

Partially supported by the United States-Israel Binational Science Foundation. I thank Alice Leonhardt for the excellent typing. Done 10/94, (set theoretic, pseudo descriptive set theory) 22-29/11/94 (essentially §1-§2); more 9-14/12/95 (fulfilling §1,§3 eliminating use of the maximal model in λ^{+2}, (new §4) of categoricity in λ^{+2} (see §7,§8), more 18-2 of 4/95 (redo §3,§10). §10 written 1/2/95 First typed at Rutgers - 95/Feb/10 Latest Revision 99/Apr/15

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Annotated Content

§0 \underline{Introduction}

[We give three versions of the main theorem in 0.2 - 0.3. In 0.4 - 0.31 we review the relevant knowledge of abstract elementary classes to help make this paper self-contained. This includes the representation by PC-classes defined by omission of quantifier free types (0.12, 0.13); types and stability (based on ≤_{𝔎}); and the equivalence of saturation to model homogeneity (0.25).]

§1 \underline{Weak Diamond}

[We mainly present necessary material on the weak diamond, a combinatorial principle whose main variant holds for λ^{+} if 2^{λ} < 2^{λ^{+}}. We state cases provable from ZFC together with suitable cardinal arithmetic assumptions (1.2). We present applications of weak diamond to the number of models of fixed cardinality (notably in 1.10). We deal also with the definable weak diamond, and introduce a smaller ideal of “small” sets: UDmId^{ℱ} (λ) (1.14 - 1.17).]

§2 \underline{First Attempts}

[We define the class K_{λ}^{3} of triples (M, N, a) representing types in 𝒮 (M) for M ∈ K_{λ}, and start to investigate it, dealing with the weak extension property, the extension property, minimality, reduced types (except for minimality, in the first order case, these hold trivially). Our aims are to have the extension property or at least the weak extension property for all triples in K_{λ}^{3}, and the density of minimal triples. The first property makes the model theory more like the first order case, and the second is connected with categoricity. We start by proving the weak extension property under reasonable assumptions. We prove the density of minimal triples under the strong assumption K_{λ^{+3}} = ∅ and an extra cardinal arithmetic assumption (2^{λ^{+}} > λ^{++}). In the end, under the additional assumption K_{λ^{+3}} = 0 we prove that all triples have the extension property and that we have disjoint amalgamation in K_{λ}. Now the assumption K_{λ^{+3}} = 0 does no harm if we just want to prove Theorem 0.2. The reader willing to accept these assumptions may skip some proofs later. The proof of the extension property makes essential use of categoricity in λ^{+}.]

§3 \underline{Non-structure}

[We try to present clearly and in some generality the construction of many models in λ^{++} based on knowledge of models of size λ^{+}, using weak diamond on λ^{+} and on λ^{++}. This is done by forming a tree ⟨M̄^{η} : η ∈ ^{λ^{++}>}2⟩ with M̄^{η} an ≤_{𝔎}-increasing continuous sequence of members of K_{λ} with limit ⋃_{i<λ^{+}} M_{i}^{η} increasing with η (and an additional restriction). Actually λ^{+}

can be replaced by a regular uncountable λ′ (so ∥M_{i}^{η}∥ = λ is replaced by ∥M_{i}^{η}∥ < λ′)].

§4 \underline{Minimal types} [We prove that every member of K_{λ}^{3} has the extension property, by proving it for minimal triples. We use: if M_{ℓ} ∈ K_{λ} and for every minimal p ∈ 𝒮 (M_{0}) the set 𝒮_{≥p}(M_{1}) has cardinality ≤ λ^{+}, then the M ∈ K_{λ^{+}} is saturated for minimal types and hence the number of minimal types in 𝒮 (M_{1}) is ≤ λ^{+} (for M_{1} ∈ K_{λ}), which is a step toward stability in λ.]

§5 \underline{Inevitable types and stability in λ} [We continue to “climb the ladder”, using the amount of structure we already have (and sometimes categoricity) to get more. We start by assuming there are minimal types, and show that some minimal types are inevitable, construct p_{i} ∈ 𝒮 (N_{i}) minimal (i ≤ λ^{+}) both strictly increasing continuous and with p_{0}, p_{δ} inevitable, and then as in the proof of the equivalence of saturativity and model homogeneity, we show N_{δ} is universal over N_{0}. We can then deduce stability in λ, so the model in λ^{+} is saturated. Then we note that we have disjoint amalgamation in K_{λ}.]

§6 \underline{A proof for 𝔎 categorical in λ^{+2}} [We give a shortcut to proving the main theorem by using stronger assumptions. If I(λ^{+2}, K) = 1 and I(λ^{+3}, K) = 0 then in some triple (M, N, a) ∈ K_{λ^{+}}, a is essentially algebraic over M, i.e. this is a maximal triple. Now first assuming for some pair M_{0} ≤_{𝔎} M_{2} in K_{λ} we have unique amalgamation for every possible M_{1} with M_{0} ≤_{𝔎} M_{1} ∈ K_{λ} (and using stability), we get a pair of models in λ^{+} which contradicts the existence of maximal triples. We then use the methods of §3 to prove that there are enough cases of unique amalgamation.]

§7 \underline{Extensions and Conjugacy} [We investigate types. We prove that in 𝒮 (N), N ∈ K_{λ} the following reduced implies inevitable, and that non-algebraic extensions preserve the conjugacy classes (so solving the realize/ materialize problem). Hence if ⟨N_{i} : i < α⟩ is <_{𝔎}-increasing in K_{λ} and λ divides α then N_{α} is (λ, cf(α))- saturated over N_{0}.]

§8 \underline{Uniqueness of Amalgamation in 𝔎_{λ}} [We have by §6 that some pairs M_{0}^{∗} <_{𝔎} M_{2}^{∗} in 𝔎_{λ} satisfy M_{0}^{∗} ≤_{𝔎} M_{1} ⇒ M_{0}^{∗}, M_{1}, M_{2}^{∗} has unique (disjoint) amalgamation. Now if we have a ≤_{𝔎}- increasing continuous sequence ⟨N_{i} : i ≤ α⟩ such that (N_{i}, N_{i+1}) ≅ (M_{0}^{∗}, M_{2}^{∗}), we can amalgamate N_{0}, M_{1}, N_{α} whenever N_{0} ≤_{𝔎} M_{1}, step by step. So some uniqueness is preserved and N_{α} can be any (λ, cf(α))-saturated model over N_{0}. When we require also saturativity of M_{i} and of the resulting model, we get a nonforking relation denoted NF_{λ,δ̄}. We define the general nonforking relation NF_{λ} by closing NF_{λ,δ̄} downward. So we succeed to define a relation which should behave as a nice nonforking relation. But we have to work to prove that this relation satisfies the expected properties, first for the “saturated” version and then in the general case by a diagram chase.]

§9 \underline{Nice extensions in K_{λ^{+}}} [As we have a notion of “nonforking” amalgamation in K_{λ}, we can use it to build ≤_{𝔎}-extensions M_{1} ∈ K_{λ^{+}} for any given M_{0} ∈ K_{λ^{+}}. This defines naturally a two-place relation ≤_{λ^{+}}^{∗} on K_{λ^{+}}; “being a nice ≤_{𝔎}-submodel”. We investigate it and variants. In particular, we prove the existence of disjoint amalgamation.]

§10 \underline{Non-structure for ≤_{λ^{+}}^{∗}} [Instead proving that all disjoint amalgamations in K_{λ} are nonforking ones, we prove that on K_{λ^{+}} the relation <_{λ^{+}}^{∗} is the same as ≤_{λ}^{∗}, which is just as good for our purpose. Toward this we assume a failure and get many pairwise non-isomorphic models in K_{λ^{+2}}, contradicting an assumption of 0.2(2). But once we have that ≤_{𝔎} agrees on K_{λ^{+}} with ≤_{λ^{+}}^{∗} we have disjoint amalgamation in K_{λ^{+}}, which suffices for building a model in K_{λ^{+3}}.]

§0 Introduction

Makowski [Mw85] is a readable and good exposition concerning categoricity in abstract elementary classes around ℵ_{1}. Our primary concern is:

Problem 0.1. Can we have some (not necessarily much) classification theory for reasonable non-first order classes 𝔎 of models, with no uses of even traces of compactness and only mild set theoretic assumptions?

Let me try to clarify the meaning of Problem 0.1. What is the meaning of “\underline{mild set theoretic assumptions}?” We are allowing requirements on cardinal arithmetic like GCH and weaker relatives. Preferably, assumptions like diamonds and squares and even mild large cardinals will not be used (apart from cases provable in ZFC, or in ZFC plus allowable assumptions).

In fact we try to continue [Sh 88], where results about the number of nonisomorphic models in ℵ_{1} and ℵ_{2} of a sentence ψ ∈ L_{ω_{1},ω} are obtained. Now in [Sh 88] the theorem parallel to the present one is proved assuming 2^{ℵ_{0}} < 2^{ℵ_{1}}, so it is quite natural to use such assumptions here.

What is the meaning of “some classification theory?” While the dream is to have a classification theory as “full” as the one obtained in [Sh:c], we will be glad to have theorems speaking just on having few models in some cardinals or even categoricity and at least one in others. E.g. by [Sh 88] if ψ ∈ L_{ω_{1},ω} satisfies 1 ≤ I(ℵ_{1}, ψ) < 2^{ℵ_{1}} (and 2^{ℵ_{0}} < 2^{ℵ_{1}}) then I(ℵ_{2}, ψ) > 0. Here I(μ, 𝔎) is the number of models in 𝔎 of cardinality μ, up to isomorphism.

What are “reasonable non first order classes?” This means we allow classes of “locally finite” or “atomic” structures, or structures “omitting a type”, or more generally the class of models of a sentence in L_{κ,ω}, (i.e. allowing conjunction < κ but quantification only over a finite string) but not one restricting ourselves to e.g. well orderings. In fact, we use “abstract elementary classes” from [Sh 88] (reviewed below).

What is the meaning of “uses traces of compactness?” For non first order classes we cannot use the powerful compactness theorem, but there are many ways to get weak forms of it: one way is using large cardinals (compact cardinals in Makkai Shelah [MaSh 285], or just measurable cardinals as in Kolman Shelah [KlSh 362], or in [Sh 472]). Another way is to use “non-definability of well ordering” which follows from the existence of Ehrenfeucht-Mostowski models, and also from ψ ∈ L_{ω_{1},ω} having uncountable models (used extensively in [Sh 88]). Our aim is to use \underline{none of this} and we would like to see if any theory is left.

Above all, we hope the proofs will initiate classification theory in this case, so we hope the flavour will be one of introducing and investigating notions of a model theoretic character. Proofs of, say, a descriptive set theory character, will not satisfy this hope.

It seems to us that this goal is met here. We prove (see 6.12):

Theorem 0.2. (2^{λ} < 2^{λ^{+}} < 2^{λ^{++}}). Let 𝔎 be an abstract elementary class. If 𝔎 categorical in λ, λ^{+} and λ^{++} \underline{then} I(λ^{+3}, 𝔎) > 0.

Here λ^{+3} = λ^{+++} and in general ℵ_{α}^{+β} means ℵ_{α+β}. Of course, the categoricity in three successive cardinals is a strong assumption. Now note that in [Sh 88], the categoricity in ℵ_{0} is gained “freely”, so the gap is smaller than seems at first glance. Still it is better to have

Theorem 0.3. If 𝔎 is categorical in λ and λ^{+} with λ > ℵ_{0} and 1 ≤ I(λ^{++}, 𝔎) < μ_{wd}(λ^{++}), \underline{then} I(λ^{+3}, 𝔎) > 0.

Note however:

(α) A silly point; at exactly one point in the proof of 0.3 we assume λ > ℵ_{0} (in the proof of 4.6). This is silly as our intent is to prove for general λ what we know for λ = ℵ_{0} by [Sh 88]; however, there we assume 𝔎 is PC_{ℵ_{1},ℵ_{0}}, a reasonable assumption, but one which is not assumed here. We shall complete this in [Sh 600, §2], so we do not mention the assumption λ > ℵ_{0} in theorems relying on 4.6.

(β) More seriously, at some point we assume toward a contradiction that K_{λ^{+3}} = ∅ in order to prove the density of the set of minimal triples. This is fine for proving theorem 0.2, but is not desirable if we want to develop a classification theory. This will also be dealt with in [Sh 600, §2].

(γ) Concerning μ_{wd}(λ^{+2}): in 6.12 we get I(λ^{++}, K) ≥ μ_{wd}(λ^{++}) instead of I(λ^{++}, K) = 2^{λ^{++}}. The cardinal μ_{wd}(χ) is defined in 1.1(6). It is almost 2^{χ}; i.e. always 2^{χ} < μ_{wd}(χ^{+}) ≤ 2^{χ^{+}} ≤ μ_{wd}(χ^{+})^{ℵ_{0}}. Already in [Sh 87b] sometimes instead of getting 2^{χ} we get only μ_{wd}(χ) (again see [Sh 600]). If you are satisfied with I(χ, K) ≥ μ_{wd}(χ) in those cases, you can ignore some proofs.

We present below, as background, the following open questions which appeared in [Sh 88], for 𝔎 an abstract elementary class, of course, e.g. the class of models of ψ ∈ L_{κ^{+},ω} with the relation M ≤_{𝔎} N being M ≺_{ℒ} N for ℒ a fragment of L_{κ^{+},ω} to which ψ belongs. In [Sh 87a], [Sh 87b], [Sh 88] we prove:

(∗)_{3} categoricity (of ψ ∈ L_{ω_{1},ω}(Q)) in ℵ_{1} implies the existence of a model of ψ

of cardinality ℵ_{2}; (∗)_{4} if n > 0, 2^{ℵ_{0}} < 2^{ℵ_{1}} < ⋯ < 2^{ℵ_{n}}, ψ ∈ L_{ω_{1},ω} and 1 ≤ I(ℵ_{ℓ}, ψ) < μ_{wd}(ℵ_{ℓ}) for

1 ≤ ℓ ≤ n, \underline{then} ψ has a model of cardinality ℵ_{n+1}.

Now the problems were: \underline{Problem} 1) Prove (∗)_{3}, (∗)_{4} in the context of an abstract elementary class 𝔎 which is P C_{ℵ_{0}}.

\underline{Problem} 2) Parallel results in ZFC; e.g. prove (∗)_{3} when n = 1, 2^{ℵ_{0}} = 2^{ℵ_{1}}.

\underline{Problem} 3) Construct examples; e.g. 𝔎 (or ψ ∈ L_{ω_{1},ω}), categorical in ℵ_{0}, ℵ_{1}, …, ℵ_{n} but not in ℵ_{n+1}.

\underline{Problem} 4) If 𝔎 is P C_{λ} (abstract elementary class), and is categorical in λ and λ^{+}, does it necessarily have a model in λ^{++}?

Concerning Problem 3, by Hart Shelah [HaSh 323],2.10(2) + 3.8 there is ψ_{n} ∈ ℒ_{ω_{1},ω} categorical in ℵ_{0}, ℵ_{1}, …, ℵ_{k−1}, but not categorical in λ if 2^{λ} > 2^{ℵ_{k−1}}.

The direct motivation for the present work, is that Grossberg asked me (Oct. 94) some questions in this neighborhood, in particular:

\underline{Problem} 5) Assume K = Mod(T), (i.e. K is the class of models of T), T ⊆ L_{ω_{1},ω}, |T | = λ, I(λ, K) = 1 and 1 ≤ I(λ^{+}, K) < 2^{λ^{+}}. Does it follow that I(λ^{++}, K) > 0? We think of this as a test problem and much prefer a model theoretic to a set theoretic solution. This is closely related to Problem 4 above and to [Sh 88, Theorem 3.7] (where we assume categoricity in λ^{+}, do not require 2^{λ} < 2^{λ^{+}} \underline{but} take λ = ℵ_{0} or some similar cases) and [Sh 88, Theorem 5.17(4)] (and see [Sh 88, 5.1,4.5] on the assumptions) (there we require 2^{λ} < 2^{λ^{+}}, 1 ≤ I(λ^{+}, K) < 2^{λ^{+}} \underline{and} λ = ℵ_{0}). As said above, we are dealing with a closely related problem. Problem 0.1 was stated a posteriori but is, I think, the real problem.

In a first try we used more set theory, i.e. we used the definitional weak diamond on both λ^{+} and λ^{++} (see Definition 1.13) and things like “a nice equivalence relation on 𝒫(λ) has either few or many classes” (see §1). Here we take a model theoretic approach.

We feel that this paper provides a reasonable positive solution to problem 0.1, with a classification theory flavor. We shall continue in [Sh 600] toward a parallel of [Sh 87a], [Sh 87b]. Grossberg and Shelah, in the mid-eighties, started to write a paper [GrSh 259] to prove that: if ψ ∈ L_{λ^{+},ω} has models of arbitrarily large cardinality, and is categorical in μ^{+n} for each n and if μ ≥ λ and 2^{μ^{+n}} < 2^{μ^{+n+1}} for n < ω, \underline{then} ψ is categorical in every μ′ ≥ μ; this is a weak form of the upward part of Los’ conjecture. See Makkai Shelah [MaSh 285] on T ⊆ ℒ_{κ,ω} where κ is a compact cardinal; where we get downward and upward theorems for successor cardinals which are sufficiently bigger than κ + |T |. On the downward part, see [KlSh 362],[Sh 472] which deals with a downward theorem for successor cardinals which are sufficiently larger than κ + |T | when the theory T is in the logic ℒ_{κ,ω} and κ is measurable. See also [Sh 394] which deals with abstract elementary classes with amalgamation, getting similar results with no large cardinals. Part of §1 and §3 have a combinatorial character. Most of the paper forms the content of a course given in Fall ’94 (essentially without §3, §9, §10). The paper is written with an eye to developing the model theory, rather than just proving Theorem 0.2.

∗  ∗  ∗

On abstract elementary classes see [Sh 88] and [Sh 300, II,§3]. To make the paper self-contained, we will review some relevant definitions and results. We thank Gregory Cherlin for improving the writing of §1-§3 and for breaking 3.25, 3.26 and 3.27 to three proofs. This work is continued in [Sh 600].

Abstract elementary classes

0.4 Conventions. 𝔎 = (K, ≤_{𝔎}), where K is a class of τ-models for a fixed vocabulary τ = τ_{K} = τ_{𝔎} and ≤_{𝔎} is a two-place relation on the models in K. We do not always strictly distinguish between K and (K, ≤_{𝔎}). We shall assume that K, ≤_{𝔎} are fixed, and M ≤_{𝔎} N ⇒ M, N ∈ K; and we assume that the following axioms hold. When we use < in the sense of elementary submodel for first order logic, we write <_{ℒ_{ω,ω}} .

0.5 Definition.

Ax 0: The validity of M ∈ K or of N ≤_{𝔎} M depends on N and M only up to isomorphism - in the second case, isomorphism of the pair.

Ax I: If M ≤_{𝔎} N then M ⊆ N (i.e. M is a submodel of N).

Ax II: ≤_{𝔎} is transitive and reflexive on K.

Ax III: If λ is a regular cardinal, M_{i} (i < λ) is ≤_{𝔎}-increasing (i.e. i < j < λ implies M_{i} ≤_{𝔎} M_{j}) and continuous (i.e. for limit ordinals δ < λ we have

M_{δ} = ⋃_{i<δ} M_{i}) \underline{then} M_{0} ≤_{𝔎} ⋃_{i<λ} M_{i}.

Ax IV: If λ is a regular cardinal, M_{i} (i < λ) is ≤_{𝔎}-increasing continuous,

M_{i} ≤_{𝔎} N \underline{then} ⋃_{i<λ} M_{i} ≤_{𝔎} N.

Ax V: If M_{0} ⊆ M_{1} and M_{ℓ} ≤_{𝔎} N for ℓ = 0, 1, then M_{0} ≤_{𝔎} M_{1}.

Ax V I: There is a cardinal λ such that: if A ⊆ N and |A| ≤ λ \underline{then} for some M ≤_{𝔎} N we have A ⊆ |M| and ∥M∥ ≤ λ. We define the Löwenheim-Skolem number LS(𝔎) as the least such λ with λ ≥ |τ|.

\underline{Notation}: K_{λ} = \{M ∈ K : ∥M∥ = λ\} and K_{<λ} = ⋃_{μ<λ} K_{μ}.

ℒ_{ω,ω} is first order logic. A theory in ℒ (τ) is a set of sentences from ℒ (τ).

0.6 Definition. The embedding f : N → M is a 𝔎-embedding or a ≤_{𝔎}-embedding if its range is the universe of a model N′ ≤_{𝔎} M, (so f : N → N′ is an isomorphism (onto)).

Very central in [Sh 88], but peripheral here is:

0.7 Definition. 1) For a logic ℒ and vocabulary τ, ℒ (τ) is the set of ℒ -formulas in this vocabulary.

2) Let T_{1} be a theory in ℒ_{ω,ω}(τ_{1}), τ ⊆ τ_{1} vocabularies, Γ a set of types in ℒ_{ω,ω}(τ_{1}); (i.e. for some m, a set of formulas φ(x_{0}, …, x_{m−1}) ∈ ℒ_{ω,ω}(τ_{1})). Then we let EC(T_{1}, Γ) = {M : M a τ_{1}-model of T_{1} which omits every p ∈ Γ}.

3) P C_{τ} (T_{1}, Γ) = P C(T_{1}, Γ, τ) = {M : M is a τ-reduct of some M_{1} ∈ EC(T_{1}, Γ)}.

4) We say that 𝔎 is P C_{λ}^{μ} if for some T_{1}, T_{2}, Γ_{1}, Γ_{2} and τ_{1} and τ_{2} we have: T_{ℓ} is a first order theory in the vocabulary τ_{ℓ}, Γ_{ℓ} is a set of types in the vocabulary τ_{ℓ} and K = P C(T_{1}, Γ_{1}, τ_{𝔎}) and {(M, N) : M ≤_{𝔎} N, M, N ∈ K} = P C(T_{2}, Γ_{2}, τ′) where τ′ = τ_{𝔎} ∪ {P } (P a new one place predicate) and |T_{ℓ}| ≤ λ, |Γ_{ℓ}| ≤ μ for ℓ = 1, 2. If μ = λ, we may omit it.

0.8 Example. If τ_{1} = τ, T_{1}, Γ are as above, and (K, ≤_{𝔎}) is defined by K =: EC(T_{1}, Γ), ≤_{𝔎} =: ≺_{ℒ_{ω,ω}} , \underline{then} the pair satisfies the Axioms from 1.2 and LS(𝔎) ≤ |T_{1}| + ℵ_{0} + |τ_{1}|.

0.9 Example. (V = L).

Let cf(λ) ≥ ℵ_{1}, n < ω \underline{then} for some ψ ∈ L_{λ^{+},ω} we have: ψ has no model of cardinality λ^{+(n+1)}, and is categorical in λ^{+n} (i.e. has one and only one model up to isomorphism). Let M^{∗} = (L_{λ^{+n}}, ∈, i)_{i≤λ} and let ψ be

⋀{φ : φ is a first order sentence which M^{∗} satisfies}

∧ (∀x) (x ∈ λ ≡ ⋁_{i<λ} x = i) .

0.10 Lemma. Let I be a directed set (i.e. partially ordered by ≤, such that any two elements have a common upper bound).

1) If M_{t} is defined for t ∈ I, and t ≤ s ∈ I implies M_{t} ≤_{𝔎} M_{s} \underline{then} for every t ∈ I

we have M_{t} ≤_{𝔎} ⋃_{s∈I} M_{s}.

2) If in addition t ∈ I implies M_{t} ≤_{𝔎} N \underline{then} ⋃_{s∈I} M_{s} ≤_{𝔎} N.

Proof. By induction on |I| (simultaneously for 1) and 2)), or see [Sh 88, 1.6]. □_{0.10}

0.11 Lemma.

1) Let τ_{1} = τ ∪ {F_{i}^{n} : i < LS(𝔎) and n < ω}, F_{i}^{n} an n-place function symbol (assuming, of course, F_{i}^{n} ∉ τ). If M_{1} is an expansion of M to a τ_{1}-model and ā ∈ n|M| for some n, let Mā be the submodel of M with universe {F_{i}^{n}(ā) : i < LS(𝔎)}. Every model M from K can be expanded to a τ_{1}-model M_{1} such that:

(A) Mā ≤_{𝔎} M for any ā ∈ ^{n}|M|; (B) if ā ∈ ^{n}|M| then ∥Mā∥ ≤ LS(𝔎); (C) if b̄ is a subsequence of ā (even up to rearrangement), \underline{then} Mb̄ ≤_{𝔎} Mā; (D) for every N_{1} ⊆ M_{1} we have N_{1} ↾ τ ≤_{𝔎} M.

2) If N ≤_{𝔎} M is given, then we can choose the expansion M_{1} so that clauses (A)

- (D) hold and

(E) N = N_{1} ↾ τ for some N_{1} ⊆ M_{1}.

Proof. We define by induction on n, the values of F_{i}^{n}(ā) for every i < LS(𝔎), ā ∈ ^{n}|M| such that ā ⊆ N ⇒ Mā ⊆ N when we are proving (2). By Ax V I there is an Mā ≤_{𝔎} M, ∥Mā∥ ≤ LS(𝔎) such that |Mā| includes

⋃{Mb̄ : b̄ a subsequence of ā of length < n} ∪ ā

and Mā does not depend on the order of ā. Let |Mā| = {c_{i} : i < i_{0} ≤ LS(𝔎)}, and define F_{i}^{n}(ā) = c_{i} for i < i_{0} and F_{i}^{n}(ā) = c_{0} for i_{0} ≤ i < LS(𝔎) (so we can demand “F_{i}^{n} is symmetric”).

Clearly our conditions are satisfied: if b̄ is a subsequence of ā then Mb̄ ≤_{𝔎} Mā by Ax V. □_{0.11}

0.12 Lemma. 1) There is a set Γ of types in ℒ_{ω,ω}(τ_{1}) (where τ_{1} is as in Lemma 0.11) such that K = P C(∅, Γ, τ). So 𝔎 is a P C_{LS(𝔎)}^{2^{LS(𝔎)}}-class, see Definition 0.7(4). The types above consist of quantifier-free formulas.

2) Moreover, if N_{1} ⊆ M_{1} ∈ EC(∅, Γ), and N, M are the τ-reducts of N_{1}, M_{1} respectively \underline{then} N ≤_{𝔎} M. Also, if N ≤_{𝔎} M \underline{then} there is an τ_{1}-expansion M_{1} of M and a submodel N_{1} of M_{1} such that M_{1} ∈ EC(∅, Γ) and N_{1} ↾ τ = N.

3) Also {(M, N) : N ≤_{𝔎} M} is a P C_{LS(𝔎)}^{(2^{LS(𝔎)})}-class, hence 𝔎 is as well.

Proof. 1) Let Γ_{n} be the set of complete quantifier free n-types p(x_{0}, …, x_{n−1}) in ℒ_{ω,ω}(τ_{1}) such that: if M_{1} is an L_{1}-model, ā realizes p in M_{1} and M is the Lreduct of M_{1}, then Mb̄ ≤_{𝔎} Mā ∈ 𝔎 for any subsequence of b̄ of ā. Recall that Mc̄ (for c̄ ∈ ^{m}|M_{1}|) is the submodel of M whose universe is {F_{i}^{m}(c̄) : i < LS(𝔎)} (and there are such submodels) and subsequence include permutation.

Let Γ be the set of p which are complete quantifier free n-types for some n < ω in ℒ_{ω,ω}(τ_{1}) and which do not belong to Γ_{n}. So if M^{1} is in P C(∅, Γ, τ_{1}) then by 0.10 we have M_{1} ↾ τ ∈ K hence P C(∅, Γ, L) ⊆ K and by 0.11(1) we have K ⊆ P C(∅, Γ, L).

2) The first phrase is proven as in (1). For the second phrase, use 0.11(2).

3) Follows from (2). □_{0.12}

0.13 Conclusion. There is τ_{1} with τ ⊆ τ_{1} and |τ_{1}| ≤ LS(𝔎) such that: for any M ∈ K and any τ_{1}-expansion M_{1} of M which is in EC(∅, Γ),

N_{1} ⊆ M_{1} ⇒ N_{1} ↾ τ ≤_{𝔎} M
N_{1} ⊆ N_{2} ⊆ M_{1} ⇒ N_{1} ↾ τ ≤_{𝔎} N_{2} ↾ τ.

0.14 Conclusion. If for every α < (2^{LS(𝔎})^{+}, K has a model of cardinality ≥ ℶ_{α} \underline{then} K has a model in every cardinality ≥ LS(𝔎).

Proof. Use 0.12 and the value of the Hanf number for: models of a first order theory omitting a given set of types, for languages of cardinality LS(K) (see [Sh:c, VII,§5] and history there).

□_{0.14}

0.15 Definition. For λ regular > LS(𝔎) and N ∈ 𝔎_{λ} we say N̄ = ⟨N_{α} : α < λ⟩ is a representation of N if N̄ is ≤_{𝔎}-increasing continuous, ∥N_{α}∥ < λ and N = ⋃_{α<λ} N_{α}.

If λ = μ^{+} then if not said otherwise, we require ∥N_{α}∥ = μ.

How will we define types, and in particular, the set 𝒮 (M) of complete types over M, when no formulas are present? If we have a “monster model” ℭ we can use automorphisms; but any such “monster” is far down the road. So we will “chase diagrams” in K_{λ} (being careful, not to use excessivley large models). This gives us a relation of “having the same type” we call E_{μ}^{at}, but this relation in general is not transitive (if we do not have amalgamation in K_{λ}). So E_{μ} will be defined as the transitive closure of E_{μ}^{at}.

0.16 Definition. 1) The two-place relation E_{M}^{at} is defined on triples (M, N, a) with M fixed, M ≤_{𝔎} N ∈ K_{∥M∥}, and a ∈ N by:

(M, N_{1}, a_{1})E_{M}^{at}(M, N_{2}, a_{2}) \underline{iff} there is N ∈ K_{μ} and ≤_{𝔎} -embeddings
f_{ℓ} : N_{ℓ} → N for ℓ = 1, 2 such that:
f_{1} ↾ M = id_{M} = f_{2} ↾ M and f_{1}(a_{1}) = f_{2}(a_{2}).

Let E_{M} be the transitive closure of E_{M}^{at}.

2) For μ ≥ LS(𝔎) and M ∈ K_{μ} we define 𝒮 (M) as {tp(a, M, N) : M ≤_{𝔎} N ∈ K_{μ} and a ∈ N} where tp(a, M, N) = (M, N, a)/E_{M}.

3) We say “a realizes p in N” if a ∈ N, p ∈ 𝒮 (M) and for some N′ ∈ K_{μ} we have M ≤_{𝔎} N′ ≤_{𝔎} N, a ∈ N′, and p = tp(a, M, N′).

4) We say “a_{2} strongly realizes (M, N^{1}, a^{1})/E_{M}^{at} in N” if for some N^{2}, a^{2} we have M ≤_{𝔎} N^{2} ≤_{𝔎} N, a_{2} ∈ N^{2}, and (M, N^{1}, a^{1}) E_{M}^{at} (M, N^{2}, a^{2}).

5) We say M_{0} ∈ 𝔎 is an amalgamation base if letting λ = ∥M_{0}∥ we have: for every M_{1}, M_{2} ∈ 𝔎_{λ} and ≤_{𝔎}-embeddings f_{ℓ} : M_{0} → M_{ℓ} (for ℓ = 1, 2) there is M_{3} ∈ 𝔎_{λ} and ≤_{𝔎}-embeddings g_{ℓ} : M_{ℓ} → M_{3} (for ℓ = 1, 2) such that g_{1} ∘ f_{1} = g_{2} ∘ f_{2}.

6) We say 𝔎 is stable in λ \underline{if} LS(𝔎) ≤ λ and for all M ∈ K_{λ} we have |𝒮 (M)| ≤ λ.

0.17 Observation. If M is an amalgamation base then E_{M} = E_{M}^{at}, and we have:

“a strongly realizes (M, N, b)/E_{M} in N” iff “a realizes (M, N, b)/E_{M}”.

0.18 Definition.

1) 𝔎 has the λ-amalgamation property or has amalgamation in λ, if every M_{0} ∈ 𝔎_{λ} is an amalgamation base (see 0.16(5) above).

2) N is universal over M if: M ≤_{𝔎} N and if M ≤_{𝔎} N′ ∈ K_{≤∥N∥} then N′ can be ≤_{𝔎}-embedded into N over M.

3) 𝔎 has universal extensions in λ \underline{if} for every M ∈ K_{λ} there is N such that:

(a) M ≤_{𝔎} N ∈ K_{λ}

(b) N is universal over M.

4) N_{1}, N_{2} have a joint embedding if for some N ∈ K there are ≤_{𝔎}-embeddings h_{ℓ} of N_{ℓ} into N for ℓ = 1, 2. Let JEP_{μ} (JEP) means this holds for N_{1}, N_{2} in K_{μ} (in K).

5) 𝔎_{λ} has unique amalgamation (or 𝔎 has unique amalgamation in λ) when: if M_{i}^{ℓ} ∈ K_{λ} for ℓ < 2, i < 4, and for i = 1, 2 we have h_{i,0}^{ℓ} is a ≤_{𝔎}-embedding of M_{0}^{ℓ} into M_{i}^{ℓ}, h_{3,i}^{ℓ} is a ≤_{𝔎}-embedding of M_{i}^{ℓ} into M_{3}^{ℓ}, h_{3,1}^{ℓ} ∘ h_{1,0}^{ℓ} = h_{3,2}^{ℓ} ∘ h_{2,0}^{ℓ} and Rang(h_{3,1}^{ℓ}) ∩ Rang(h_{3,2}^{ℓ}) = Rang(h_{3,1}^{ℓ} ∘ h_{1,0}^{ℓ}) and for i < 3, f_{i} is an isomorphism from M_{i}^{0} onto M_{i}^{1}, f_{0} ⊆ f_{1}, f_{0} ⊆ f_{2} \underline{then} for some N ∈ K_{λ} there are 𝔎-embedding h_{ℓ} : M_{3}^{ℓ} → N such that h_{0} ∘ h_{3,i}^{0} = h_{1} ∘ h_{3,i}^{1}, for i = 1, 2.

6) Let p_{ℓ} ∈ 𝒮 (M_{ℓ}) for ℓ = 0, 1. We say p_{0} ≤ p_{1} \underline{if} M_{0} ≤_{𝔎} M_{1} and for some N and a we have M_{1} ≤_{𝔎} N ∈ K_{∥M_{1}∥+LS(𝔎)}, a ∈ N and tp(a, M_{ℓ}, N) = p_{ℓ} for ℓ = 0, 1. We also write p_{0} = p_{1} ↾ M_{0} (p_{0} is unique knowing M_{0}, M_{1}, p_{1} hence p_{1} ↾ M_{0} is well defined).

0.19 Claim.

1) If 𝔎 is categorical in λ and LS(𝔎) ≤ λ then: there is a model in K_{λ^{+}} \underline{iff} for some (equivalently, every) model M ∈ K_{λ} there is N such that M <_{𝔎} N ∈ K_{λ} \underline{iff} for some (equivalently every) M ∈ K_{λ} there is N such that M <_{𝔎} N ∈ K_{λ^{+}}.

2) If 𝔎 has amalgamation in λ, LS(𝔎) ≤ λ, and M_{0} ≤_{𝔎} M_{1} are in K_{λ} with M_{0} ≤_{𝔎} N_{0} ∈ K_{λ^{+}} \underline{then} we can find h and N_{1} such that N_{0} ≤_{𝔎} N_{1} ∈ K_{λ^{+}} and h is a ≤_{𝔎}-embedding of M_{1} into N_{1} extending id_{M_{0}}. We can allow N_{0} ∈ K_{μ} with μ > λ if 𝔎 has amalgamation in every λ′ ∈ [λ, μ).

3) Assume 𝔎 has amalgamation in λ and LS(𝔎) ≤ λ. If M_{0} ≤ M_{1} are from K_{λ} and p_{0} ∈ 𝒮 (M_{0}) \underline{then} we can find an extension p_{1} ∈ 𝒮 (M_{1}) of p_{0}.

Proof. 1) As then we can choose by induction on i < λ^{+} models M_{i} ∈ K_{λ}, ≤_{𝔎}-increasing continuous, M_{i} ≠ M_{i+1}, for i = 0 use K_{λ^{+}} ≠ ∅, for i limit take union, for i = j + 1 use the previous sentence; so M_{λ^{+}} = ∪{M_{i} : i < λ^{+}} ∈ K_{λ^{+}} as required. 2), 3) Left to the reader. □_{0.19}

0.20 Remark. We can here add the content of 6.5, 6.7, 6.12.

0.21 Definition. 1) For λ > LS(𝔎) we say “N ∈ 𝔎 is λ-saturated” \underline{if} for every M ≤_{𝔎} N of cardinality < λ, if M ≤_{𝔎} N′ ∈ K_{<λ} and a′ ∈ N′ then some a ∈ N strongly realizes (M, N′, a′)/E_{M}^{at} (in the interesting cases this suffices).

2) We say “N ∈ 𝔎 is λ-saturated above μ (or is λ-saturated ≥ μ)” \underline{if} above we restrict ourselves to M of cardinality ≥ μ. (Cf. (λ, κ)-saturated in Definition 0.27, 0.28 below).

0.22 Fact. 1) If LS(𝔎) ≤ μ < λ, λ is regular, 𝔎 has the amalgamation property in every μ′ ∈ [μ, λ), and for all M ∈ K_{[μ,λ)} we have |𝒮 (M)| ≤ λ and λ = cf(λ), \underline{then} there is some M ∈ K_{λ} saturated above μ.

2) Assume λ > μ ≥ LS(𝔎) and N ∈ K and 𝔎 has amalgamation in K_{μ_{1}}, for every μ_{1} ∈ [μ, λ). \underline{Then}: N is λ-saturated above μ \underline{iff} for every M ≤_{𝔎} N of cardinality < λ but ≥ μ, every p ∈ 𝒮 (M) is realized in N (i.e. for some a ∈ N we have tp(a, M, N) = p).

3) If LS(𝔎) ≤ μ_{0} ≤ μ′_{0} ≤ μ′ ≤ μ and M is μ-saturated above μ_{0}, \underline{then} it is μ′- saturated above μ′_{0}. If LS(𝔎) ≤ μ_{0} < μ \underline{then}: M is μ-saturated above μ_{0} \underline{iff} for every λ ∈ [μ_{0}, μ), M is λ^{+}-saturated above λ.

Proof. Check.

0.23 Definition. The type p ∈ 𝒮 (M) is local when: for any directed partial order

I and models M_{t} ≤_{𝔎} M for t ∈ I with I ⊧ t ≤ s ⇒ M_{t} ≤_{𝔎} M_{s} and M = ⋃_{t∈I} M_{t},

and any p′ ∈ 𝒮 (M) \underline{if} (p ↾ M_{t} = p′ ↾ M_{t}) for all t ∈ I \underline{then} p = p′. We say M is local if every p ∈ 𝒮 (M) is, and 𝔎 is local if every M ∈ 𝔎 is. We can add “above μ” as in Definition 0.21(2).

0.24 Definition. 1) I(λ, K) = I(λ, 𝔎) is the number of M ∈ K_{λ} up to isomorphism.

2) 𝔎 (or K) is categorical in λ if I(λ, K) = 1.

3) IE(λ, 𝔎) = sup{|K′| : K′ ⊆ K_{λ} and for M ≠ N in K′_{λ}, M is not ≤_{𝔎}-embeddable into N}. Abusing notation, if we write IE(λ, K) ≥ μ, we mean that for some K′ ⊆ K_{λ} as above, |K′| ≥ μ, and similarly for = μ. If there is a problem with attainment of the supremum we shall say explicitly.

0.25 The Model-homogeneity = Saturativity Lemma. Let λ > μ ≥ LS(𝔎) and M ∈ K.

1) M is λ-saturated above μ \underline{iff} M is (𝒟_{𝔎}, λ)-homogeneous above μ, which means: for every N_{1} ≤_{𝔎} N_{2} ∈ K such that μ ≤ ∥N_{1}∥ ≤ ∥N_{2}∥ < λ and N_{1} ≤_{𝔎} M there is a ≤_{𝔎}-embedding f of N_{2} into M over N_{1}.

2) If M_{1}, M_{2} ∈ K_{λ} are λ-saturated above μ < λ and for some N_{1} ≤_{𝔎} M_{1}, N_{2} ≤_{𝔎} M_{2}, both of cardinality ∈ [μ, λ), we have N_{1} ≅ N_{2} \underline{then} M_{1} ≅ M_{2}; in fact, any isomorphism f from N_{1} onto N_{2} can be extended to an isomorphism from M_{1} onto M_{2}.

3) If in (2) we demand only “M_{2} is λ-saturated” and M_{1} ∈ K_{≤λ} \underline{then} f can be extended to a ≤_{𝔎}-embedding from M_{1} into M_{2}.

4) In part (2) instead of N_{1} ≅ N_{2} it suffices to assume that N_{1} and N_{2} can be ≤_{𝔎}-embedded into some N ∈ K, which holds if 𝔎 has the JEP or just JEP_{μ}.

Proof. 1) The “if” direction is easy as λ > LS(𝔎). Let us prove the other direction, so without loss of generality ∥N_{1}∥ = μ = ∥N_{2}∥.

By 0.18(6) without loss of generality λ is regular, moreover N_{2} has cardinality ∥N_{1}∥.

Let |N_{2}| = {a_{i} : i < κ}, and we know κ = ∥N_{1}∥ = ∥N_{2}∥ < λ. We define by induction on i ≤ κ, N_{1}^{i}, N_{2}^{i}, f_{i} such that:

(a) N_{1}^{i} ≤_{𝔎} N_{2}^{i} and ∥N_{2}^{i}∥ < λ

(b) N_{1}^{i} is ≤_{𝔎}-increasing continuous in i

(c) N_{2}^{i} is ≤_{𝔎}-increasing continuous in i

(d) f_{i} is a ≤_{𝔎}-embedding of N_{1}^{i} into M (e) f_{i} is increasing continuous in i

(f) a_{i} ∈ N_{1}^{i+1}

(g) N_{1}^{0} = N_{1}, N_{2}^{0} = N_{2}, f_{0} = id_{N_{1}}

(h) N_{1}^{i} and N_{2}^{i} has cardinality κ.

For i = 0, clause (g) gives the definition. For i limit let:

N_{1}^{i} = ⋃_{j<i} N_{1}^{j} and

N_{2}^{i} = ⋃_{j<i} N_{2}^{j} and

f_{i} = ⋃_{j<i} f_{j}.

Now (a)-(f) continues to hold by continuity (and ∥N_{2}^{i}∥ < λ as λ is regular).

For i successor we use our assumption; more elaborately, let M_{1}^{i−1} ≤_{𝔎} M be f_{i−1}(N_{1}^{i−1}) and M_{2}^{i−1}, g_{i−1} be such that g_{i−1} is an isomorphism from N_{2}^{i−1} onto M_{2}^{i−1} extending f_{i−1}, so M_{1}^{i−1} ≤_{𝔎} M_{2}^{i−1} (but without loss of generality M_{2}^{i−1} ∩ M = M_{1}^{i−1}). Now apply the saturation assumption with M, M_{1}^{i−1}, tp(g_{i−1}(a_{i−1}), M_{1}^{i−1}, M_{2}^{i−1}) here standing for N, M, p there (note: a_{i−1} ∈ N_{2} = N_{2}^{0} ⊆ N_{2}^{i−1} and λ > κ = ∥N_{2}^{i−1}∥ = ∥M_{2}^{i−1}∥ and ∥M_{1}^{i−1}∥ = ∥N_{1}^{i−1}∥ ≥ ∥N_{1}^{0}∥ = ∥N_{1}∥ ≥ μ so the requirements and also the cardinality in Definition 0.21 holds). So there is b ∈ M such that tp(b, M_{1}^{i−1}, M) = tp(g_{i−1}(a_{i−1}), M_{1}^{i−1}, M_{2}^{i−1}). Moreover (if 𝔎 has amalgamation in μ the proof is slightly shorter) remembering the second sentence in 0.21(1) which speaks about “strongly realizes” there is b ∈ M such that b strongly realizes (M_{1}^{i−1}, M_{2}^{i−1}, g_{i−1}(a_{i−1}))/E_{M_{1}^{i−1}}^{at} in M. This means

(see Definition 0.16 (3)) that for some M_{1}^{i,∗} we have M_{1}^{i−1} ≤_{𝔎} M_{1}^{i,∗} ≤_{𝔎} M and (M_{1}^{i−1}, M_{2}^{i−1}, g_{i−1}(a_{i−1}))E_{M_{1}^{i−1}}^{at}(M_{1}^{i−1}, M_{1}^{i,∗}, b). This means (see Definition 0.16(1)

that M_{1}^{i,∗} too has cardinality κ and there is M_{i}^{2,∗} ∈ K_{κ} such that M_{1}^{i−1} ≤_{𝔎} M_{2}^{i,∗} and there are ≤_{𝔎}-embeddings h_{2}^{i}, h_{1}^{i} of M_{2}^{i−1}, M_{1}^{i,∗} into M_{2}^{i,∗} over M_{1}^{i−1} respectively, such that h_{2}^{i}(g_{i−1}(a_{i−1})) = h_{1}^{i}(b). Now changing names, without loss of generality h_{2}^{i} is the identity. Let N_{2}^{i}, h_{i} be such that N_{2}^{i−1} <_{𝔎} N_{2}^{i}, h_{i} an isomorphism from N_{2}^{i} onto M_{2}^{i,∗} extending g_{i−1}. Let N_{1}^{i} = h_{i}^{−1} ∘ h_{1}^{i}(M_{1}^{i,∗}), f_{i} = (h_{1}^{i})^{−1} ∘ (h_{i} ↾ N_{1}^{i}).

We have carried the induction. Now f_{κ} is a ≤_{𝔎}- embedding of N_{1}^{κ} into M over N_{1}, but |N_{2}| = {a_{i} : i < κ} ⊆ N_{1}^{κ}, so f_{κ} ↾ N_{2} : N_{2} → M is as required. 2), 3) By the hence and forth argument (or see [Sh 300, II,§3] = [Sh:h, II,§3]).

4) Easy, too. □_{0.25}

0.26 Remark. Note that by 0.25(2) if M is μ-saturated above μ_{0} and 𝔎 has the JEP_{μ_{0}} then 𝔎 has λ-amalgamation for each λ ∈ [μ_{0}, μ).

0.27 Definition. Fix λ ≥ κ with κ regular.

1) We say that N_{1} ∈ K_{λ} is (λ, κ)-saturated over N_{0} or that (N_{1}, c)_{c∈N_{0}} is (λ, κ)- saturated (and λ-saturated of cofinality κ means (λ, κ)-saturated) if: there is a sequence ⟨M_{i} : i < κ⟩ which is ≤_{𝔎}-increasing continuous with M_{0} = N, M_{κ} = N_{1} and M_{i+1} ∈ K_{λ} universal over M_{i} (see 0.18(2)).

2) If we omit κ, we mean κ = cf(λ); (λ, α)-saturated means (λ, cf(α))-saturated; and N_{1} is (λ, 1)-saturated over N_{0} means just N_{0} ≤_{𝔎} N_{1} are in K_{λ}.

0.28 Claim. Fix λ ≥ κ with κ regular.

1) If N_{1}, N_{2} are (λ, κ)-saturated over N, \underline{then} N_{1}, N_{2} are isomorphic over N (i.e. the (λ, κ)-saturated model over N ∈ K_{λ} is unique over N).

2) If K_{λ} ≠ ∅, λ ≥ κ = cf(κ) and over every M ∈ K_{λ} there is N with M ≤_{𝔎} N ∈ K_{λ} universal over M, \underline{then} for every N ∈ K_{λ} there is N_{1} ∈ K_{λ} which is (λ, κ)-saturated over N.

3) If 𝔎_{λ} has amalgamation and 𝔎 is stable in λ (i.e. M ∈ K_{λ} ⇒ |𝒮 (M)| ≤ λ) \underline{then} every M ∈ K_{λ} has a universal extension (so part (2)’s conclusion holds).

Proof. See [Sh 300, Ch.II] or check. □_{0.28}

We do not need at present but recall from [Sh 88]:

0.29 Claim. There is τ′ ⊇ τ ∪ {P_{0}, P_{1}, P_{2}, c} of cardinality ≤ LS(𝔎) with c an individual constant, with P_{ℓ} unary predicates, and a set Γ of quantifier free types such that:

(a) if M′ ∈ P C_{τ′}(∅, Γ) and M_{ℓ} = (M′ ↾ τ) ↾ P_{ℓ}^{M′} for ℓ = 0, 1, 2, \underline{then} M_{ℓ} ∈ K, M_{0} ≤_{𝔎} M_{1}, M_{0} ≤_{𝔎} M_{2}, c^{M′} ∈ M_{2}, and N′ ⊆ M′ ⇒ (N′ ↾ τ) ↾ P_{ℓ}^{N′} ≤_{𝔎} M_{ℓ}, and there is no b ∈ M_{1} satisfying:

⊗ for every ā ∈ ^{ω>}(P_{0}^{M′}), letting Nā be the τ′-submodel of M′ generated by ā and M ℓ ; ā = M_{ℓ} ↾ (M_{ℓ} ∩ Nā), we have M ℓ ; ā ≤_{𝔎} M_{ℓ}, and b strongly

realizes tp(c^{M′}, M 0 ; ā, M 2 ; ā) in M 1 ; ā

(b) if M_{ℓ}^{∗} for ℓ = 0, 1, 2 and c are as in (a) and M_{0} = M_{1} ∩ M_{2}, \underline{then} for some M′ we have clause (a).

Proof. Should be clear; see [Sh 88]. □_{0.29}

0.30 Remark.

1) Claim 0.29 enables us to translate results of the form: the existence of a two cardinal with omitting types model in λ_{2} implies the existence of one in λ_{1}, provided that types are local in the sense that p ∈ 𝒮 (M) is determined by ⟨p ↾ N : N ≤_{𝔎} M, ∥N∥ ≤ λ⟩.

2) This enables us to prove implications between cases of λ-categoricity, \underline{if} we have a nice enough theory of types as in [Sh:c, VIII,§4]; if we have in λ_{2} a saturated model, categoricity in λ_{1} implies categoricity in λ_{2}. Also (if we know a little more) categoricity in λ_{2} is equivalent to the non-existence of a non-saturated model in λ_{2}.

0.31 Claim. 1) Assume M_{n} ≤_{𝔎} M_{n+1}, M_{n} ∈ K_{λ}, 𝔎 has amalgamation in λ. If p_{n} ∈ 𝒮 (ℳ_{n}), p_{n} ≤ p_{n+1} (i.e. p_{n} = p_{n+1} ↾ M_{n}, see 0.18(6)), \underline{then} there is

p ∈ 𝒮 ( ⋃_{n<ω} M_{n}) such that n < ω ⇒ p_{n} ≤ p.

2) If ⟨M_{i} : i ≤ δ⟩ is ≤_{𝔎}-increasing continuous, p_{i} ∈ 𝒮 (M_{i}), (j < i ⇒ p_{j} ≤ p_{i}), p_{i} = tp(a_{i}, M_{i}, N_{i}) and h_{i,j} is a ≤_{𝔎}-embedding of N_{j} into N_{i} (for j < i < δ) such that h_{i,j} ↾ M_{j} = id_{M_{j}}, h_{i,j}(a_{j}) = a_{i}, \underline{then} there is p_{δ} ∈ 𝒮 (M_{δ}), i ≤ δ ⇒ p_{i} ≤ p_{δ}.

3) If 𝔎 has amalgamation in λ and is stable in λ (i.e. M ∈ K_{λ} ⇒ |𝒮 (M)| ≤ λ), \underline{then}

(a) every M ∈ K_{λ} has a universal extension;

(b) for every M ∈ K_{λ} and regular θ ≤ λ there is N ∈ K_{λ} which is (λ, θ)- saturated over M.

§1 Weak Diamond

1.1 Definition. Fix λ regular and uncountable.

1) WDmTId(λ, S, χ̄) = {A :A ⊆ ∏_{α∈S} χ_{α}, and for some function F with domain
⋃_{α<λ}^{α}(2^{<λ}) mapping ^{α}(2^{<λ}) into χ_{α},
for every η ∈ A, for some f ∈ ^{λ}(2^{<λ}) the set
{δ ∈ S : η(δ) = F (f ↾ δ)} is not stationary}.

(Note: WDmTId stand for weak diamond target ideal)^{1}. Here we can replace 2^{<λ} by any set of this cardinality, and so we can replace f ∈ ^{λ}(2^{<λ}) by f_{1}, …, f_{n} ∈ ^{λ}(2^{<λ}) with F being n-place.

2) cov_{wdmt}(λ, S, χ̄) = Min{|𝒫| : 𝒫 ⊆ WDmTId(λ, S) and ^{S}2 ⊆ ⋃_{A∈𝒫} A}

3) WDmTId_{<μ}(λ, S, χ̄) = {A : for some i^{∗} < μ and A_{i} ∈ WDmTId(λ, S) for

i < i^{∗} we have A ⊆ ⋃_{i<i^{∗}} A_{i}}

4)  WDmId_{<μ}(λ, χ̄) = {S ⊆ λ : cov_{wdmt}(λ, S) < μ}.

5) Instead of “< μ^{+}” we may write μ, if we omit μ we mean (2^{<λ}). If χ̄ is constantly 2 we may omit it, if χ_{α} = 2^{|α|} we may write pow instead of χ̄.

6)  Let μ_{wd}(λ, χ̄) = cov_{wdmt}(λ, λ, χ̄).

7) We say that the weak diamond holds on λ if λ ∉ WDmId(λ). We may omit χ̄ when it is constantly 2.

By [DvSh 65], [Sh:b, XIV,1.18(2),1.8] (presented better in [Sh:f, AP,§1], note: 1.2(4) below rely on [Sh 460]) we have:

1in [Sh:b, AP,§1], [Sh:f, AP,§1] we express cov_{wdmt}(λ, S) > μ^{∗} by allowing f(0) ∈ μ^{∗} < μ

1.2 Theorem.

1) If λ = ℵ_{1}, 2^{ℵ_{0}} < 2^{ℵ_{1}}, μ ≤ (2^{ℵ_{0}}) or even 2^{θ} = 2^{<λ} < 2^{λ}, μ = (2^{θ})^{+}, or just: for some θ, 2^{θ} = 2^{<λ} < 2^{λ}, μ ≤ 2^{λ}, and χ^{<λ} < μ for χ < μ, \underline{then} λ ∉ WDmId_{<μ}(λ). If in addition (∗)_{<μ,λ} below holds, \underline{then} λ ∉ WDmId_{<μ}(λ, pow), where:

(∗)_{μ,λ} there are no A_{i} ∈ [μ]^{λ^{+}} for i < 2^{λ} such that i ≠ j ⇒ |A_{i} ∩ A_{j}| < ℵ_{0}

and (∗)_{<μ,λ} means (∗)_{χ,λ} holds for χ < μ.

2) If μ ≤ λ^{+} or cf([μ_{1}]^{≤λ}, ⊆) < μ for μ_{1} < μ or μ = ℵ_{0} \underline{then} WDmId_{<μ}(λ, χ̄) is a normal ideal on λ. If this ideal is not trivial, \underline{then} λ = cf(λ) > ℵ_{0}, 2^{<λ} < 2^{λ}.

3) A sufficient condition for (∗)_{<μ,λ} is:

(a) μ ≤ 2^{λ} & (∀α < μ)(|α|^{ℵ_{0}} < 2^{λ}).

4) Another sufficient condition for (∗)_{<μ,λ} is:

(b) μ ≤ 2^{λ} & λ ≥ ℶ_{ω}.

1.3 Remark. 1) So if cf(2^{λ}) < μ (which holds if 2^{λ} is singular and μ = 2^{λ}) then (∗)_{<μ,λ} implies that there is A ⊆ ^{λ}2, |A| < 2^{λ}, A ∉ WDmTId(λ).

2) Some related definitions appear in 1.13; we use them below (mainly DfWD_{<μ}(λ)), but as in a first reading it is recommended to ignore them, the definition is given later.

3) We did not look again at the case (∀σ < λ)(2^{σ} < 2^{<λ} < 2^{λ}).

As in [Sh 88, 3.5], ([Sh 87a, 2.7], [Sh 87b, 6.3]):

1.4 Claim. Assume λ ∉ WDmId_{<μ}(λ) or at least DfWD_{<μ}(λ) (where λ = cf(λ) > ℵ_{0}) and 𝔎 is an abstract elementary class.

1) Assume 𝔎 is categorical in χ, λ = χ^{+}, and 𝔎 has a model in λ (if LS(𝔎) ≤ χ this is equivalent to: the model M ∈ K_{χ} is not ≤_{𝔎}-maximal). Assume further 𝔎 does not have the λ-amalgamation property in χ. \underline{Then} for any M_{i} ∈ 𝔎_{λ} for i < i^{∗} < μ, there is N ∈ 𝔎_{λ} not ≤_{𝔎}-embeddable into any M_{i} (and the assumptions of part (2) below holds).

2) Assume M_{η} ∈ K_{<λ} for η ∈ ^{λ>}2, M_{η} = ⋃_{α<ℓg(η)} M_{η↾(α+1)}, ν ◁ η ⇒ M_{ν} ≤_{𝔎} M_{η},

and M_{ηˆ ⟨0⟩}, M_{ηˆ ⟨1⟩} cannot be amalgamated over M_{η} (hence M_{η} ≠ M_{ηˆ ⟨ℓ⟩}). Set M_{η} =: ⋃_{α<λ} M_{η↾α} for η ∈ ^{λ}2. Clearly M_{η} belongs to K_{λ}. For the DfWD_{<μ}(λ)

version assume also

(∗) ⟨M_{η} : η ∈ ^{λ>}2⟩ is definable (even just by ℒ_{λ,λ}) in 𝔅 = (ℋ (χ), ∈, <_{χ}^{∗}, 𝔎_{<χ}, λ, μ).

\underline{Then} for any N_{i} ∈ 𝔎_{λ} for i < i^{∗} < μ, there is η ∈ ^{λ}2 such that: M_{η} is not ≤_{𝔎}-embeddable into any N_{i}.

3) In part (2), if LS(𝔎) ≤ λ we can allow N_{i} ∈ K_{κ_{i}} if ∑_{i<i^{∗}} cf([κ_{i}]^{λ}, ⊆) < μ.

4) If we use DfWD_{<μ}(λ): assume only λ ∉ WDmId_{<μ}(λ, χ̄). Part (2) holds if M_{η}

is defined for η ∈ ⋃_{α<λ} ∏_{i<α} χ_{i}, and ε < ζ < χ_{i}, η ∈ ∏_{j<i} χ_{j} ⇒ M_{ηˆ ⟨ε⟩}, M_{ηˆ ⟨ζ⟩} cannot be

amalgamated over M_{η}. The assumption of Part (4) (hence the conclusion of Part (2)) holds if we assume that for M ∈ K_{χ}, i < λ there are χ_{i} ≤_{𝔎}-extensions of M in K_{λ}, which pairwise cannot be amalgamated over M.

Proof. 1) It is straightforward to choose M_{η} ∈ K_{χ} for η ∈ ^{α}2 by induction on α, as required in part (2). Then use part (2) to get the desired conclusion.

2) Without loss of generality the universe of N_{i} is λ and the universe of M_{η} is an ordinal γ_{η} such that η ∈ ^{λ>}2 ⇒ γ_{η} < λ and η ∈ ^{λ}2 ⇒ γ_{η} = λ. The reader can ignore the “DfWD_{<μ}(λ)” version (ignoring the h_{η}’s, g) if he likes. For α < λ and η ∈ ^{α}2 let the function h_{η} be h_{η}(i) = M_{η↾(i+1)} for i < ℓg(η). Let ⟨M_{η} : η ∈ ^{λ>}2⟩ be the <_{χ}^{∗}-first such object. For each i < i^{∗} we define A_{i} ⊆ ^{λ}2 by A_{i} = {η ∈ ^{λ}2 : M_{η} can be ≤_{𝔎}-embedded into N_{i}}.

For η ∈ A_{i} choose f_{η} : M_{η} → N_{i}, a ≤_{𝔎}- embedding, hence f_{η} ∈ ^{λ}λ. We also define a function F_{i} from ⋃_{α<λ}(^{α}2 ×^{α}λ) to {0, 1} by:

F_{i}(η, f) is : 0 \underline{if} f is a ≤_{𝔎} -embedding of M_{η} into N_{i}

with range ⊆ ℓg(η) which can be extended to a ≤_{𝔎} -embedding of M_{ηˆ ⟨0⟩} into N_{i}

F_{i}(η, f) is 1 otherwise.

Now for any η ∈ A_{i}, the set

E = {δ < λ : γ_{η↾δ} = δ and f_{η} ↾ δ is a function from δ to δ}

is a club of λ. For every δ ∈ E clearly F (η ↾ δ, f_{η} ↾ δ) = η(δ) (as M_{ηˆ ⟨0⟩}, M_{ηˆ ⟨1⟩} cannot be amalgamated over M_{η}).

Hence (for the “Def” version see Definition 1.13(2) using 1.14(1), 1.14(3)) we have A_{i} ∈ WDmTId^{Def}(λ). As i^{∗} < μ clearly ⋃_{i<i^{∗}} A_{i} ∈ WDmId_{<μ}^{Def}(λ) and hence

by assumption of the claim ^{λ}2 ≠ ⋃_{i} A_{i}. Take η ∈ ^{λ}2\ ⋃_{i<i^{∗}} A_{i}. Then M_{η} is as

required.

3) Without loss of generality the universe of N_{i} is κ_{i}. Let 𝒫_{i} ⊆ [κ_{i}]^{λ} be a set of minimal cardinality such that (∀B)[B ⊆ κ_{i} & |B| ≤ λ → (∃B′ ∈ 𝒫_{i})(B ⊆ B′)]. As LS(𝔎) ≤ λ we can find for each A ∈ 𝒫_{i}, a model N_{A}^{i} ≤_{𝔎} N_{i} of cardinality λ^{+} whose universe includes A. Now apply part (2) to {N_{A}^{i} : i < i^{∗} and A ∈ 𝒫_{i}}.

4) Same proof. □_{1.4}

We give three variants of the preceding:

1.5 Claim. 1) Assume

(∗)_{1} λ = cf(λ) > ℵ_{0} (a) M_{η} is a τ-model of cardinality < λ for η ∈ ^{λ>}2 and

(b) for each η ∈ ^{λ}2, ⟨M_{η↾α} : α < λ⟩ is ⊆-increasing continuous with union, called M_{η} ∈ K, of cardinality λ

(c) if η ∈ ^{λ>}2, η ◁ ρ_{ℓ} ∈ ^{λ}2 for ℓ = 1, 2 then M_{ρ_{1}}, M_{ρ_{2}} are not isomorphic over M_{η}

(∗)_{2} λ ∉ WDmId_{<μ}(λ)

\underline{Then} I(λ, K) ≥ μ, and in fact we can find X ⊆ ^{λ}2 of cardinality ≥ μ such that {M_{ρ} : ρ ∈ X} are pairwise non-isomorphic.

2) Assume

(∗)_{1}^{d} like (∗)_{1} in part (1) but in addition (d) ⟨M_{η} : η ∈ ^{λ>}2⟩ is definable in 𝔅 = 𝔅_{χ}

(∗)_{2}^{d} λ ∉ DfWD_{<μ}(λ).

\underline{Then} the conclusion of part (1) holds.

3) The parallel of 1.4(4) holds.

Proof. 1) Let {N_{i} : i < i^{∗}} be a maximal subset of {M_{ρ} : ρ ∈ ^{λ}2} consisting of pairwise non-isomorphic models, and use the proof of 1.4(2) with f_{η} : N_{i} ≃ M_{η}. 2), 3) Left to the reader. □_{1.5}

1.6 Claim. 1) Assume

(∗)_{1} M_{η} ∈ K_{<λ} for η ∈ ^{λ>}2, ⟨M_{η↾α} : α ≤ ℓg(η)⟩ is ≤_{𝔎}-increasing continuous, and M_{ηˆ ⟨1⟩} cannot be ≤_{𝔎}-embedded into M_{ν} over M_{η} when

ηˆ ⟨0⟩ ⊴ ν ∈ ^{λ>}2

(∗)_{2} λ = cf(λ) > ℵ_{0}, λ ∉ WDmId_{<μ}(λ), and λ is a successor cardinal, or at least there is no λ-saturated normal ideal on λ, or at least WDmId(λ) is not λ-saturated (which holds if for some θ < λ, {δ < λ : cf(δ) = θ} ∉ WDmId(λ)).

\underline{Then} there is A ⊆ ^{λ}2, |A| = 2^{λ} such that: if η_{1} ≠ η_{2} are in A then (taking M_{η} =

⋃_{α<λ} M_{η↾α})

(α) M_{η_{1}} ≇ M_{η_{2}} for η_{1} ≠ η_{2} ∈ A and (β) if (2^{χ})^{+} < 2^{λ} for χ < λ \underline{then} we can also achieve: M_{η_{1}} cannot be ≤_{𝔎}-

embedded into M_{η_{2}}.

2) Under the assumptions of 1.4(1) we can find ⟨M_{η} : η ∈ ^{λ>}2⟩ as in the assumption of 1.4(2).

3) Under the assumption of 1.4(2) the assumption of 1.6(1) holds.

4) Under the assumption of 1.6(1) we have I(λ, 𝔎) = 2^{λ} and if (2^{χ})^{+} < 2^{λ} then IE(λ, 𝔎) = 2^{λ}.

5) The parallel of 1.4(4) holds.

Proof of 1.6. 1) The proof of [Sh 88, 3.5] works (see the implications preceding it). More elaborately, we divide the proof into cases according to the answer to the following:

\underline{Question}: Is there η^{∗} ∈ ^{λ>}2 such that for every ν satisfying η^{∗} ⊴ ν ∈ ^{λ>}2 there are ρ_{0}, ρ_{1} ∈ ^{λ>}2 such that: ν ◁ ρ_{0}, ν ⊴ ρ_{1}, and M_{ρ_{0}}, M_{ρ_{1}} cannot be amalgamated over M_{η^{∗}}?

We can find a function h : ^{λ>}2 → ^{λ>}2, such that:

(a) the function h is one-to-one, preserving ◁ and (h(ν))ˆ ⟨ℓ⟩ ⊴ h(νˆ ⟨ℓ⟩) (b)_{yes} \underline{when} the answer to the question is yes, it is exemplified by η^{∗} = h(⟨⟩) and

M_{h(νˆ ⟨0⟩)}, M_{h(νˆ ⟨1⟩)} cannot be amalgamated over M_{η^{∗}} (for every ν ∈ ^{λ>}2) (b)_{no} when the answer to the question above is no, h(⟨⟩) = ⟨⟩, ℓ < 2 and if

νˆ ⟨ℓ⟩ ⊴ ρ_{0}, νˆ ⟨ℓ⟩ ◁ ρ_{1} then M_{h(ρ_{0})}, M_{h(ρ_{1})} can be amalgamated over M_{h(ν)}.

Without loss of generality h is the identity, by renaming (and we can preserve (∗)_{1}^{d} of 1.5(2) in the relevant case). Also clearly M_{ηˆ ⟨ℓ⟩} ≠ M_{η} (by the non-amalgamation assumption).

\underline{Case 1} The answer is yes. We do not use the non λ-saturation of WDmId(λ) in this case. For any η ∈ ^{λ}2 and ≤_{𝔎}-embedding g of M_{⟨⟩} into M_{η} =: ⋃_{α<λ} M_{η↾α}, let

A_{η,g} =: {ν ∈ ^{λ}2 : there is a ≤_{𝔎} -embedding of M_{ν} into M_{η} extending g}

A_{η} =: {ν ∈ ^{λ}2 : there is a ≤_{𝔎} -embedding of M_{ν} into M_{η}}.

So: |A_{η,g}| ≤ 1 for any g and η ∈ A_{η} (as if ν_{1}, ν_{2} ∈ A_{η,g} are distinct then for some ordinal α < λ and ν ∈ ^{α}2 we have ν =: ν_{0} ↾ α = ν_{1} ↾ α, ν_{0}(α) ≠ ν_{1}(α) and use the choice of h(νˆ ⟨ℓ⟩)). Since A_{η} = ∪{A_{ν,g} : g is a ≤_{𝔎} -embedding of M_{⟨⟩} into M_{η}}, we have |A_{η}| ≤ 2^{<λ}. Hence we can choose by induction on ζ < 2^{λ}, η_{ζ} ∈ ^{λ}2\ ⋃_{ξ<ζ} A_{η_{ξ}} (existing by cardi-

nality considerations as 2^{<λ} < λ). Then ξ < ζ ⇒ M_{η_{ξ}} ≇ M_{η_{ζ}} so we have proved clause (α). If (2^{<λ})^{+} < 2^{λ}, then use the Hajnal free subset theorem ([Ha61]) to choose distinct {η_{ζ} : ζ < 2^{λ}} ⊆ ^{λ}2 such that ζ ≠ ξ ⇒ η_{ζ} ∉ A_{η_{ξ}}, so we have proved clause (β), too.

\underline{Case 2}: The answer is no. Again, without loss of generality M_{η} has as universe the ordinal γ_{η}. Let ⟨S_{i} : i < λ⟩ be a partition of λ to sets, none of which is in WDmId(λ). For each i we define a function F_{i} as follows:

if δ ∈ S_{i}, η, ν ∈ ^{δ}2, γ_{η} = γ_{ν} = δ, and f : δ → δ then
F_{i}(η, ν, f) = 0 if f can be extended to an embedding of
M_{νˆ ⟨0⟩} into some M_{ρ} with ηˆ ⟨0⟩ ⊴ ρ.
F_{i}(η, ν, f) = 1 otherwise.

So as S_{i} ∉ WDmId(λ), for some η_{i}^{∗} ∈ ^{λ}2 we have:

(∗) for every η ∈ ^{λ}2, ν ∈ ^{λ}2 and f ∈ ^{λ}λ the following set of ordinals i < λ is stationary:

{δ ∈ S_{i} : F_{i}(η ↾ δ, ν ↾ δ, f ↾ δ) = η_{i}^{∗}(δ)}.

Now for any X ⊆ λ let η_{X}, ρ_{X} ∈ ^{λ}2 be defined by:

if α ∈ S_{i} then i ∈ X ⇒ η_{X}(α) = 1 − η_{i}^{∗}(α), i ∉ X ⇒ η_{X}(α) = 0 and
ρ_{X} = η_{{2i:i∈X}∪{2i+1:i∉X}}.

Now we show

(∗) if X, Y ⊆ λ, and X ≠ Y then M_{ρ_{X}} cannot be ≤_{𝔎}-embedded into M_{ρ_{Y}}.

Clearly (∗) will suffice for finishing the proof.

Assume toward a contradiction that f is a ≤_{𝔎}-embedding of M_{ρ_{X}} into M_{ρ_{Y}}; as X ≠ Y there is i such that i ∈ X ⇔ i ∉ Y so there is j ∈ {2i, 2i + 1} such that ρ_{X} ↾ S_{j} = ⟨1 − η_{j}^{∗}(α) : α ∈ S_{j}⟩ and ρ_{Y} ↾ S_{j} is identically zero. Clearly E = {δ : f maps δ into δ} is a club of λ and hence S_{j} ∩ E ≠ ∅.

So if δ ∈ S_{j} ∩ E then f ↾ M_{(ρ_{Y} ↾(δ+1))} = f ↾ M_{(ρ_{Y} ↾δ)ˆ ⟨0⟩} extend f ↾ M_{ρ_{Y} ↾δ} and is a ≤_{𝔎}-embedding of it into some M_{(ρ_{X}↾α)}, α < λ large enough.

Now by the choice of h we get

δ ∈ S_{j} ∩ E ⇒ F (ρ_{X} ↾ δ, ρ_{Y} ↾ δ, f ↾ δ) = ρ_{X}(δ) = 1 − η_{j}^{∗}(δ).

But this contradicts the choice of η_{j}^{∗}.

2) This corresponds to Case I, where we do not need the weak diamond, but just 2^{<λ} < 2^{λ}. 3),4),5). Check, similarly. □_{1.6}

1.7 Conclusion. 1) Assume

(∗)_{1} for η ∈ ^{λ>}2, M_{η} ∈ K_{<λ} and ⟨M_{η↾α} : α ≤ ℓg(η)⟩ is ≤_{𝔎}-increasing continuous and M_{ηˆ ⟨1⟩} cannot be ≤_{𝔎}-embedded into M_{ν} over M_{η} when ηˆ ⟨0⟩ ⊴ ν ∈ ^{λ>}2

and ⟨M_{η} : η ∈ ^{λ>}2⟩ is definable in 𝔅

(∗)_{2}^{d} WDmId^{Def}(λ) or DfWD^{+}(λ) is not λ-saturated (which holds if there is no normal λ-saturated ideals on λ (which holds for non Mahlo λ) and holds if for some θ, {δ < λ : cf(δ) = θ} is not in the ideal).

\underline{Then} the conclusion of 1.6 holds.

From the Definition below, we use here mainly “superlimit”

1.8 Definition. 1) M ∈ 𝔎_{λ} is a superlimit if

(a) for every N ∈ 𝔎_{λ} such that M ≤_{𝔎} N there is M′ ∈ K_{λ} such that N ≤_{𝔎} M′, N ≠ M′, and M ≅ M′;

(b) if δ < λ^{+} is limit, ⟨M_{i} : i < δ⟩ is ≤_{𝔎}-increasing and M_{i} ≅ M (for i < δ)

then ⋃_{i<δ} M_{i} ≅ M.

2) For Θ ⊆ {μ : ℵ_{0} ≤ μ ≤ λ, μ regular} we say M ∈ 𝔎_{λ} is a (λ, Θ)-superlimit if:

(a) clause (α) from part (1) holds and

(b) if M_{i} ≅ M is (≤_{𝔎})-increasing for i < μ ∈ Θ then ⋃_{i<μ} M_{i} ≅ M.

3) For S ⊂ λ^{+} we say M ∈ 𝔎_{λ} is a (λ, S)-strong limit if:

(a) clause (a) from part (1) holds

(b) there is a function F from ⋃_{α<λ^{+}} ^{α}(K_{λ}) to K_{λ} such that:

(α) for any sequence ⟨M_{i} : i < α⟩ if α < λ^{+}, M_{0} = M, M_{i} is ≤_{𝔎}-increasing,

and M_{i} ∈ 𝔎_{λ}, \underline{then} j < α ⇒ M_{j} ≤_{𝔎} F (⟨M_{i} : i < α⟩)

(β) if ⟨M_{i} : i < λ^{+}⟩ is ≤_{𝔎}-increasing, M_{0} = M, M_{i} ∈ 𝔎_{λ}, and for

i < κ, M_{i+1} ≤_{𝔎} F (⟨M_{j} : j ≤ i + 1⟩) ≤_{𝔎} M_{i+2} \underline{then}
{δ ∈ S| ⋃_{i<δ} M_{i} ≇ M} is not stationary.

4) M is a (λ, κ)-limit if there is a function F as in 3b(α) such that:

(a) if ⟨M_{i} : i < κ⟩ is a <_{𝔎}-increasing continuous sequence in K_{λ},

F (M̄ ↾ (i + 1)) ≤_{𝔎} M_{i+1} \underline{then} ⋃_{i<κ} M_{i} ≅ M

(b) there is at least one such sequence.

5) M is a (λ, κ)-superlimit is defined similarly, but with F omitted and M_{i+1} ≅ M.

1.9 Claim. 1) In 1.4(1) we can replace the categoricity of 𝔎 in χ by “𝔎 has a super limit model in χ” which is not an amalgamation base (see Definition 1.8). In this case the assumption of 1.4(2), and of 1.6(1) holds.

2) We can weaken (in 1.6(5)) the existence of superlimit to “for some κ = cf(κ) ≤ χ there is a (χ, {κ})-super limit model which is not an amalgamation base”; provided that we add {δ < λ : cf(δ) = κ} ∉ WDmId_{<μ}(λ) (but for 2.9(1) we need now “WDmId_{<μ}(λ) + {δ < λ : cf(δ) = κ} is not λ-saturated”. If there is S ⊆ {δ < λ : cf(δ) = κ}, which belongs to I[λ] but not to WDmId_{<μ}(λ) we can weaken the model theoretic requirement to: there is a (χ, {κ})-medium limit (see [Sh 88, Definition §3]) but not used here.

1.10 Claim. Assume 2^{λ} < 2^{λ^{+}}.

0) If 𝔎 (an abstract elementary class) is categorical in λ, LS(𝔎) ≤ λ, I(λ^{+}, 𝔎) < 2^{λ^{+}}, \underline{then} 𝔎_{λ} has amalgamation.

1) If 𝔎 (an abstract elementary class) is categorical in λ and 1 ≤ IE(λ^{+}, K) < 2^{λ^{+}} but K_{λ^{++}} = ∅ \underline{then} 𝔎 has amalgamation in λ.

2) Assume 𝔎 has amalgamation in λ, LS(𝔎) ≤ λ, K_{λ^{+}} ≠ ∅ and K_{λ^{++}} = ∅. \underline{Then} there is M ∈ K_{λ^{+}} saturated above λ.

3) If M is μ-saturated above λ, LS(𝔎) ≤ λ_{0} < λ and 𝔎 has amalgamation in every λ′_{0} ∈ [λ_{0}, λ) \underline{then} M is μ-saturated above λ_{0}.

1.11 Remark. If I(λ^{+}, 𝔎) < 2^{λ^{+}}, then the assumption 𝔎_{λ^{++}} = ∅ is not used in part (1) of 1.10; this is 1.6(1) + (2). Also if (2^{λ})^{+} < 2^{λ^{+}} then the assumption K_{λ^{++}} = ∅ is not needed in part (1) of 1.10; by 1.6(1) + (2) (note (b) of 1.6(1)).

Proof. 0) By 1.6(1) applied to λ^{+}.

1) If not, we can choose for η ∈ ^{λ^{+}>}2 a model M_{η} ∈ 𝔎_{λ} such that [ν ◁ η ⇒ M_{ν} ≤_{𝔎} M_{η}], and M_{ηˆ ⟨0⟩}, M_{ηˆ ⟨1⟩} cannot be amalgamated over M_{η}. If (2^{λ})^{+} < 2^{λ^{+}} we are done by 1.6(1), so assume (2^{λ})^{+} = 2^{λ^{+}}. For each η ∈ ^{λ^{+}}2 let M_{η} =: ⋃_{α<λ^{+}} M_{η↾α}, and let N_{η} ∈ 𝔎_{λ^{+}} be such that

M_{η} ≤_{𝔎} N_{η}, N_{η} is ≤_{𝔎}-maximal (exists as 𝔎_{λ^{++}} = ∅). Now we choose by induction ζ < (2^{λ})^{+}, η_{ζ} ∈ ^{λ^{+}}2 such that M_{η_{ζ}} is not ≤_{𝔎}-embeddable into N_{η_{ξ}} for ξ < ζ (exists by 1.4(2)). So necessarily for ξ < ζ, N_{η_{ζ}} is not ≤_{𝔎}-embeddable into N_{η_{ξ}} (as M_{η_{ζ}} ≤ N_{η_{ζ}}). Also, for ξ < ζ, N_{η_{ξ}} is not ≤_{𝔎}-embeddable into N_{η_{ζ}} as otherwise by the maximality of N_{ξ} this implies N_{ξ} ≅ N_{ζ}. So {N_{ζ} : ζ < 2^{λ^{+}}} exemplifies IE(λ^{+}, 𝔎) = 2^{λ^{+}}, contradicting an assumption.

2) A maximal model in K_{λ^{+}} will do.

3) Easy. □_{1.10}

∗  ∗  ∗

\underline{1.12 Discussion} Instead of Weak Diamond we now discuss Definable Weak Diamond, which is weaker but suffices. Compare [MkSh 313], where many Cohen subsets are added to λ and a combinatorial principle about amalgamation of configurations ⟨M_{s} : s ⊆ n, s ≠ n⟩ is gotten.

We are interested here in the case n = 1 (ordinary amalgamation); in §3, also n = 2. Even more definability can be required.

This is particularly interesting when we look at results under some other set theory, when combining 2^{λ} = 2^{λ^{+}} with definable weak diamond on λ^{+} is helpful. This played a major role in the preliminary form of this work.

1.13 Definition. 1) In Definition 1.1 we add the superscript ℱ if we restrict ourselves to functions F ∈ ℱ.

2) Fix a model 𝔅 whose universe includes λ and has a definable pairing function on λ, and a logic ℒ closed under first order operations and substitution; also allow “M ∈ 𝔎” and “M ≤_{𝔎} N” in the formulas, if it is not said otherwise. Let

ℱ_{𝔅,ℒ}^{Def} = {F : for some g ∈ ^{λ}λ and h̄ = ⟨h_{η} : η ∈ ^{λ>}2⟩ where

h_{η} : ℓg(η) → λ and h_{η↾β} ⊆ h_{η} for β < ℓg(η)

and for some sequence ψ̄ = ⟨ψ_{α} : α < λ⟩, with ψ_{α} ∈ ℒ for α < λ the following holds for every α < λ and

f ∈ ^{α}(2^{<λ}) : F (f) = 1 iff (𝔅, α, g, h_{f}) ⊧ ψ_{α} iff F (f) ≠ 0}.

3) The version of weak diamond from 1.1, restricted to the class ℱ of 1.13(2) is called the (𝔅, ℒ)-definitional version. If ℒ is ℒ_{λ,λ} we may omit it. If 𝔅 has the form (ℋ (χ), ∈, <_{χ}^{∗}, λ) we write ℱ_{ℒ}^{Def(χ)} or ℱ^{Def(χ)} instead of ℱ_{𝔅,ℒ}^{Def} or ℱ_{𝔅}^{Def} respectively. If we omit χ, we mean χ = (2^{λ})^{+} and we may put Def(χ) or Def instead of ℱ^{Def(χ)} or ℱ^{Def} in the superscript. Having the definitional version or the definable weak diamond for λ means λ ∉ WDmId^{Def}(λ).

4) Let DfWD_{<μ}(λ) mean that with 𝔅 = (ℋ (χ), ∈, <_{χ}^{∗}, λ, μ) we have λ ∉ WDmId_{<μ}^{Def}(λ). Instead of “< μ^{+}” we write “μ” and instead of “2^{<λ}” we may write nothing.

5) Let DfWD_{<μ}^{+}(λ) mean DfWD_{<μ}(λ) together with the principle ⊗_{λ} below; we adopt the same conventions as in (4) concerning μ:

⨂λ if for η ∈ ^{λ>}2, M_{η} is a τ_{𝔎}-model of cardinality < λ, ⟨M_{η↾α} : α ≤ ℓg(η)⟩ is ≤-increasing continuous, for η ∈ ^{λ}2 we let M_{η} = ⋃_{α<λ} M_{η↾α} and for

η ≠ ν ∈ ^{λ}2, M_{η} and M_{ν} are not isomorphic over M_{⟨⟩}, \underline{then} \{M_{η}/ ≅: η ∈ ^{λ}2\} has cardinality 2^{λ} (note that 2^{θ} = 2^{<λ} < 2^{λ} implies that).

1.14 Claim.

1) Assume ℒ first order or at least is definable enriching first order.

In the definition of ℱ_{𝔅,ℒ}^{Def} , we can replace “for every α < λ” by “for a club of α < λ”. In the definition of ℱ^{Def} we can let g ∈ ^{λ}(^{λ>}2) and h_{η} : ℓg(η) → ^{λ>}2. In any case WDmId_{<μ}^{ℱ}(λ) increases with ℱ and is ⊆ WDmId_{<μ}(λ), similarly for WDmTId_{<μ}^{ℱ}(λ).

2) If ℱ = ℱ_{𝔅}^{Def} and cf(μ) > λ \underline{then} WDmId_{<μ}(λ) is a normal ideal (but possibly is equal to 𝒫(λ)).

3) Assume V ⊧ “λ = χ^{+}, χ^{<χ} = χ, μ > λ” and P is the forcing notion of adding μ Cohen subsets to χ (i.e. {g : g a partial function from μ to {0} with domain of cardinality < χ}). \underline{Then} in V^{P} we have WDmId_{<μ}^{Def}(λ) is the ideal of non-stationary subsets of λ; i.e. with 𝔅 = (ℋ (χ), ∈, <_{μ}^{∗})^{V^{P}} for \underline{any} χ. Also ⊗_{λ} of Definition 1.13(5) holds.

Remark. In 1.14(1) we use the assumption on ℒ; anyhow not serious: reread the definition 1.1(1).

Proof. 1); 2) By manipulating the h’s (using the pairing function on λ).

3) See [MkSh 313] or think (the point being that we can break the forcing, first adding ψ̄ and g (or the < μ ones) and then (read 1.1(1)) choose η ∈ ^{λ}2 as g_{∼} ↾ [γ, γ + λ) not “used before”. Now for any candidate f ∈ ^{λ}(^{λ>}2) for a club of

δ < λ, η(δ) = g(γ + δ) is not used in the definition of f ↾ δ, h_{f} ↾ δ so stationarily often η(δ) “guesses” rightly.) □_{1.14}

1.15 Claim. 1) If 2^{θ} = 2^{<λ} < 2^{λ} then DfWD^{+}(λ).

2) DfWD_{<μ}(λ) holds when λ ∉ WDmId_{<μ}(λ) (see 1.2 for sufficient conditions).

\underline{1.16 Discussion}: We hope to get successful “guessing” not just on a stationary set, but on a positive set for the same ideal for which have guessed; i.e. there is I a normal ideal on λ such that for A ∈ I^{+} there is η ∈ ^{ζ}2 guessing I-positively; this is connected to questions on λ^{+}-saturation. For more see [Sh 638].

We phrased the following notion originally in the hope of later eliminating μ_{wd}(λ) (i.e. using 2^{λ} instead of μ_{wd}(λ)).

1.17 Definition. 1)

UDmId_{<μ}^{ℱ}(λ) = {S ⊆ λ : for some i^{∗} < μ and F_{i} ∈ ℱ (for i < i^{∗})

for every η ∈ ^{S}2 there are f ∈ ^{λ}(2^{<λ}) and i < i^{∗} and club E of λ such that: for every δ ∈ E we have: δ ∈ S ⇒ η(δ) = F_{i}(f ↾ δ)

δ ∈ λ\S ⇒ 0 = F_{i}(f ↾ δ)}.

2) We omit μ if μ = 1.

3) BA^{ℱ} (λ) is defined as the family of S ⊆ λ such that for some F ∈ ℱ and η = O_{S} the condition above holds.

1.18 Claim. Assume ℱ = ℱ_{𝔅}^{Def}.

1) In the definition 1.17(1) we can replace f ∈ ^{λ}(2^{<λ}) by f ∈ ^{λ}2 or f ∈ ^{λ}(^{λ>}2).

2) UDmId^{ℱ} (λ) is a normal ideal on λ (but possibly is 𝒫(λ)).

3) BA^{ℱ} (λ) is a Boolean algebra of subsets of λ including all non-stationary subsets of λ and even UDmId^{ℱ} (λ), and is closed under unions of < λ sets and even under diagonal union.

4) If S ∈ BA^{ℱ} (λ) and F ∈ ℱ then for some η ∈ ^{S}2 we have:

(∗) for every f ∈ ^{λ}(^{λ>}2) we have

{δ ∈ S : η(δ) = F (f ↾ δ)} ≠ ∅ mod UDmId^{ℱ} (λ).

CATEGORICITY IN TWO SUCCESSIVE CARDINALS 27

5) UDmId_{<μ}^{ℱ}(λ) ⊆ WDmId_{<μ}^{ℱ}(λ) ⊆ WDmId_{<μ}(λ) and they increase with ℱ and λ ∈ UDmId_{<μ}^{ℱ}(λ) ⇔ λ ∈ WDmId_{<μ}^{ℱ}(λ).

Proof. Straightforward. □_{1.18}

\underline{1.19 Discussion} Remember

(∗)_{1} If V = “χ = χ^{<χ} & 2^{χ} = χ^{+}”, P is the forcing notion of adding μ > χ^{+} Cohen subsets to χ \underline{then} in V^{χ}, any equivalence relation on 𝒫(λ) definable with parameters X ⊆ χ and ordinals which has at least χ^{++} equivalence classes has at least μ equivalence classes

and (see [Sh:f, XVI,§2])

(∗)_{2} ZFC is consistent with CH+ for some stationary, costationary S ⊆ ω_{1} we have

(a) WDmId(ℵ_{1}) = {A ⊆ ω_{1} : A\S is not stationary}

(b) 𝒟_{ω_{1}} + S is ℵ_{2}-saturated

and (see [Sh 587]):

(∗)_{3} ZF C + GCH is consistent with {δ < ℵ_{2} : cf(δ) = ℵ_{1}} ∈ WDmId(ℵ_{2}) and (see [Sh 208]):

(∗)_{4} ZF C + 2^{ℵ_{1}} < 2^{ℵ_{2}} is consistent with {δ < ℵ_{2} : cf(δ) = ℵ_{0}} ∈ WDmId(ℵ_{2}).

§2 First attempts

Given amalgamation in K_{λ} (cf. 2.2) we try to define and analyze types p ∈ 𝒮 (M) for M ∈ K_{λ}. But types here (as in [Sh 300]) are not sets of formulas. They may instead be represented by triples (M, N, a) with M ≤_{𝔎} N and a ∈ N\M. We look for nice types (i.e. triples) and try to prove mainly the density of the set of minimal types).

To simplify matters we allow uses of stronger assumptions than are ultimately desired (e.g. 2^{λ^{+}} > λ^{++} and/or K_{λ^{+3}} = ∅). These will later be eliminated. However the first extra assumption is still a “mild set theoretic assumption”, and the second is harmless if we think only of proving our main theorem 0.2 and not on subsequent continuations.

So the aim of this section is to show that we can start to analyze such classes.

2.1 Hypothesis. 𝔎 is an abstract elementary class.

2.2 Claim. Assume

(∗)_{λ}^{2} K is categorical in λ; 1 ≤ I(λ^{+}, K) < 2^{λ^{+}}; LS(𝔎) ≤ λ and: 2^{λ} < 2^{λ^{+}}, or at least the definable weak diamond holds for λ^{+} holds.

\underline{Then}

1) 𝔎_{λ} has amalgamation.

2) If I(λ^{++}, K) = 0 \underline{then} 𝔎 has a model in λ^{+} which is universal homogeneous above λ, hence saturated above λ (see 0.21(2)).

3) If I(λ^{++}, K) = 0 then M ∈ K_{λ} ⇒ |𝒮 (M)| ≤ λ^{+}.

Proof. 1) If amalgamation fails in K_{λ} and 2^{λ} < 2^{λ^{+}}, \underline{then} the assumptions of 1.4(1) hold with λ^{+} in place of λ. Hence by 1.6(2) the statement (∗)_{1} of 1.6(1) (see there) holds and easily also (∗)_{2} of 1.6(1), hence by 1.6(4) we have I(λ^{+}, K) = 2^{λ^{+}}, a contradiction. If 2^{λ} = 2^{λ^{+}}, we are using the variants from 1.13.

2) As I(λ^{++}, K) = 0 < I(λ^{+}, K), there is M ∈ K_{λ^{+}} which is maximal. If M is not universal homogeneous above λ then there are N_{0}, N_{1} ∈ K_{λ} with N_{0} ≤_{𝔎} M and N_{0} ≤_{𝔎} N_{1} such that N_{1} cannot be ≤_{𝔎}-embedded into M over N_{0}. Use 0.19(2) to get a contradiction.

3) Follows from (2). □_{2.2}

2.3 Definition. 1)(a) K_{λ}^{3} = {(M_{0}, M_{1}, a) : M_{0} ≤_{𝔎} M_{1} are both in K_{λ} and a ∈ M_{1}\M_{0}}.

(b)  (M_{0}, M_{1}, a) ≤ (M′_{0}, M′_{1}, a′) \underline{if} a = a′, M_{0} ≤_{𝔎} M′_{0}, M_{1} ≤_{𝔎} M′_{1}

(c)  (M_{0}, M_{1}, a) ≤_{h}(M′_{0}, M′_{1}, a′) \underline{if} h(a) = a′, and for ℓ = 0, 1 we have:
h ↾ M_{ℓ} is a ≤_{𝔎} -embedding of M_{ℓ} into M′_{ℓ}.

(d)  (M_{0}, M_{1}, a) <(M′_{0}, M′_{1}, a′) \underline{if} (M_{0}, M_{1}, a)
≤ (M′_{0}, M′_{1}, a) and M_{0} ≠ M′_{0}.

(e)  similarly <_{h}

2) (M_{0}, M_{1}, a) ∈ K_{λ}^{3} has the weak extension property \underline{if} there is (M′_{0}, M′_{1}, a) ∈ K_{λ}^{3} such that (M_{0}, M_{1}, a) ≤ (M′_{0}, M′_{1}, a) and M_{0} ≠ M_{1}.

3) (M_{0}, M_{1}, a) ∈ K_{λ}^{3} has the extension property \underline{if}: for every N_{0} ∈ 𝔎_{λ} and ≤_{𝔎}-embedding f of M_{0} into N_{0} there are N_{1}, b and g such that: (M_{0}, M_{1}, a) ≤_{g} (N_{0}, N_{1}, b) ∈ K_{λ}^{3} and g ⊇ f (so g(a) = b and g is a ≤_{𝔎}-embedding of M_{1} into N_{1}).

2.4 Claim. Assume

(∗)_{λ}^{3} LS(𝔎) ≤ λ, K is categorical in λ and in λ^{+}, and 1 ≤ I(λ^{++}, K).

\underline{Then} every (M_{0}, M_{1}, a) ∈ K_{λ}^{3} has the weak extension property that is: If M_{0} ≤_{𝔎} M_{1} are in K_{λ} and a ∈ M_{1}\M_{0}, then we can find M′_{1} in 𝔎_{λ} such that: M_{0} <_{𝔎} M_{1} hence M_{0} ≠ M′_{0} and (M_{0}, M_{1}, a) ≤ (M′_{0}, M′_{1}, a).

Proof. We can choose ⟨N_{i}, a_{i} : i < λ^{+}⟩ such that:

(a) N_{i} ∈ K_{λ} is ≤_{𝔎} -increasing continuous in i;

(b) h_{i} is an isomorphism from M_{1} onto N_{i+1} such that h_{i}(M_{0}) = N_{i}, h_{i}(a) = a_{i}.

Now as a ∈ M_{1}\M_{0} clearly i < j < λ^{+} ⇒ a_{i} ∈ N_{i+1} ≤_{𝔎} N_{j} & a_{j} ∉ N_{i+1} hence

⋃_{i<λ^{+}} N_{i} ∈ K_{λ^{+}}.

By 0.19(1) applied to λ^{+} there are M′_{0} ≤_{𝔎} M′_{1} in K_{λ^{+}}, M′_{0} ≠ M′_{1}, and there is b ∈ M′_{1}\M′_{0}. As K is categorical in λ^{+}, without loss of generality M′_{0} = ⋃_{i<λ^{+}} N′_{i}.

Let χ be large enough and 𝔅 ≺ (ℋ (χ) ∈, <_{χ}^{∗}) be such that λ ⊆ 𝔅, ∥𝔅∥ = λ and ⟨N_{i}, a_{i} : i < λ^{+}⟩, M′_{0}, M′_{1}, b and the definition of 𝔎 belong to 𝔅. Let δ = 𝔅 ∩ λ^{+}, so δ ∈ (λ, λ^{+}) is a limit ordinal and

N_{δ} ≤_{𝔎} N_{δ+1} ≤_{𝔎} M′_{1},

N_{δ} ≤_{𝔎} (M′_{1} ∩ 𝔅) ≤_{𝔎} M′_{1}

𝔅 ∩ M′_{0} = N_{δ}

N_{δ+1} ∩ (M′_{1} ∩ 𝔅) = N_{δ},

so for some N we have:

N ∈ 𝔎_{λ}, N ≤_{𝔎} M′_{1}, and (N_{δ+1} ∪ (M′_{1} ∩ 𝔅)) ⊆ N

so (see Definition 2.3(1) above)

(N_{δ}, N_{δ+1}, a_{δ}) ≤_{𝔎} (M′_{1} ∩ 𝔅, N, a_{δ}),

and b witnesses N_{δ} ≠ M′_{1} ∩ 𝔅. As (M_{0}, M_{1}, a) ≅ (N_{δ}, N_{δ+1}, a_{δ}), the result follows. □_{2.4}

2.5 Definition. 1) (M_{0}, M_{1}, a) ∈ K_{λ}^{3} is \underline{minimal} when:

if (M_{0}, M_{1}, a) ≤_{h_{ℓ}} (M′_{0}, M_{1}^{ℓ}, a_{ℓ}) ∈ K_{λ}^{3} for ℓ = 1, 2,
and h_{1} ↾ M_{0} = h_{2} ↾ M_{2}
\underline{then} tp(a_{1}, M′_{0}, M_{1}^{1}) = tp(a_{2}, M′_{0}, M_{1}^{2}).

2) (M_{0}, M_{1}, a) ∈ K_{λ}^{3} is \underline{reduced} when:

if (M_{0}, M_{1}, a) ≤ (M′_{0}, M′_{1}, a) ∈ 𝔎_{λ}^{3} then M′_{0} ∩ M_{1} = M_{0}.

3) We say p ∈ 𝒮 (M_{0}) is \underline{minimal}, where M_{0} ∈ K_{λ}, if for some a, M_{1} we have: p = tp(a, M_{0}, M_{1}) and (M_{0}, M_{1}, a) ∈ K_{λ}^{3} is minimal.

4) We say p ∈ 𝒮 (M_{0}) is \underline{reduced} where M_{0} ∈ K_{λ}, if for some a, M_{1} we have p = tp(a, M_{0}, M_{1}) and (M_{0}, M_{1}, a) ∈ K_{λ}^{3} is reduced.

2.6 Fact. 1) For every (M_{0}, M_{1}, a) ∈ K_{λ}^{3} there is a reduced (M′_{0}, M′_{1}, a) such that: (M_{0}, M_{1}, a) ≤ (M′_{0}, M′_{1}, a) ∈ K_{λ}^{3}.

2) Assume ⟨(M_{0,α}, M_{1,α}, a) : α < δ⟩ is an increasing sequence of members of K_{λ}^{3}

(a) if δ < λ^{+} \underline{then} (M_{0,α}, M_{1,α}, a) ≤ ( ⋃_{β<δ} M_{0,β}, ⋃_{β<δ} M_{1,β}, a) ∈ K_{λ}^{3}.

(b) If δ = λ^{+} the result may be in K_{λ^{+}}^{3}: if {α < δ : M_{0,α} ≠ M_{0,α+1}} is cofinal, this holds.

(c) If δ < λ^{+} and each (M_{0,α}, M_{1,α}, a) is reduced \underline{then} so is

( ⋃_{β<δ} M_{0,β}, ⋃_{β<δ} M_{1,β}, a).

3) If (M_{0}, M_{1}, a) ≤ (M′_{0}, M′_{1}, a) are in K_{λ}^{3} and the first triple is minimal \underline{then} so is the second.

4) If (M_{0}, M_{1}, a) ≤ (M′_{0}, M′_{1}, a) are in K_{λ}^{3} \underline{then} tp(a, M_{0}, M_{1}) ≤ tp(a, M′_{0}, M′_{1}); (Definition 0.23(1)).

5) If K_{λ} has amalgamation, then: (M_{0}, M_{1}, a) ∈ K_{λ}^{3} is minimal if and only if:

(∗) If (M_{0}, M_{1}, a) ≤_{h_{ℓ}} (M′_{0}, M′_{1}, a_{ℓ}) ∈ K_{λ}^{3} for ℓ = 1, 2 and h_{1} ↾ M_{0} = h_{2} ↾ M_{0} \underline{then} tp(a_{1}, M′_{0}, M′_{1}) = tp(a_{2}, M′_{0}, M′_{1}).

6) If there is no maximal member^{2} of K_{λ}^{3} and there are N_{0} <_{𝔎} N_{1} in K_{λ}, \underline{then} there are N^{0} <_{𝔎} N^{1} in 𝔎_{λ^{+}}.

7) If every triple in K_{λ}^{3} has the weak extension property, and there are N_{0} <_{𝔎} N_{1} in 𝔎_{λ} \underline{then} there are N^{0} <_{𝔎} N^{1} in 𝔎_{λ^{+}}.

8) If LS(𝔎) ≤ λ and every triple in K_{λ}^{3} has the extension property and K_{λ}^{3} ≠ ∅ \underline{then} no M ∈ K_{λ^{+}} is <_{𝔎}-maximal hence K_{λ^{++}} ≠ ∅.

9) If LS(𝔎) ≤ λ and K_{λ^{+}} ≠ ∅, \underline{then} K_{λ}^{3} ≠ ∅.

10) Assume K_{λ} has amalgamation. If (M_{0}, M_{1}, a) ∈ K_{λ}^{3} and p = tp(a, M_{0}, M) is minimal, \underline{then} (M_{0}, M_{1}, a) is a minimal triple.

Proof. Easy. Note that part (7) is 2.4.

2.7 Claim. Assume (∗)_{λ^{+}}^{2} + (∗)_{λ}^{3} (i.e. the hypothesis of 2.2 and 2.4) and 2^{λ^{+}} > λ^{++}, and K_{λ^{+3}} = ∅.

\underline{Then} in 𝔎_{λ}^{3} the minimal triples are dense (i.e. above every triple in K_{λ}^{3} there is a minimal one).

Remark. We do not intend to adopt the hypotheses “2^{λ^{+}} > λ^{++}”, K_{λ^{+3}} = ∅ indefinitely. They will be eliminated in §3.

Proof. If not, we can choose by induction on α < λ^{+}, for η ∈ ^{α}2 a triple (M_{η}^{0}, M_{η}^{1}, a_{η}) and h_{η,ν} for ν ⊴ η such that:

(i) (M_{η}^{0}, M_{η}^{1}, a_{η}) ∈ K_{λ}^{3}

(ii) ν ◁ η ⇒ (M_{ν}^{0}, M_{ν}^{1}, a_{ν}) ≤_{h_{η,ν}} (M_{η}^{0}, M_{η}^{1}, a_{η})

(iii) ν_{0} ◁ ν_{1} ◁ ν_{2} ⇒ h_{ν_{2},ν_{0}} = h_{ν_{2},ν_{1}} ∘ h_{ν_{1},ν_{0}} (iv) (M_{ηˆ ⟨ℓ⟩}^{0}, M_{ηˆ ⟨ℓ⟩}^{1}, h_{ηˆ ⟨ℓ⟩,η} ↾ M_{η}^{0}) for ℓ = 0, 1 are equal

(v) tp(a_{ηˆ ⟨0⟩}, M_{ηˆ ⟨0⟩}^{0}, M_{ηˆ ⟨0⟩}^{1}) ≠ tp(a_{ηˆ ⟨1⟩}, M_{ηˆ ⟨1⟩}^{0}, M_{ηˆ ⟨1⟩}^{1}); this makes sense as M_{ηˆ ⟨0⟩}^{0} = M_{ηˆ ⟨1⟩}^{0}

(vi) if η ∈ ^{δ}2 and δ < λ^{+} is a limit ordinal, \underline{then} M_{η}^{ℓ} = ⋃_{α<δ} h_{η,η↾α}(M_{η↾α}^{ℓ})

for ℓ = 0, 1, (vii) (M_{<>}^{0}, M_{<>}^{1}, a_{<>}) ∈ K_{λ}^{3} is a triple above which there is no minimal one.

This is straightforward: for α = 0 choose a triple in K_{λ}^{3} above which supposedly there is no minimal triple; in limit α take limits of diagrams (chasing the h’s); in successor α, use non-minimality and 2.6(5).

Let M^{∗} ∈ 𝔎_{λ^{++}} be saturated above λ (exists by 2.2(2) so it is necessarily homogeneous universal above λ^{+}, hence above λ; note: λ there stands for λ^{+} here).

We choose by induction on α < λ^{+} for η ∈ ^{α}2, a ≤_{𝔎}-embedding g_{η} of M_{η}^{0} into M^{∗} such that:

ν ◁ η ⇒ g_{ν} = g_{η} ∘ h_{η,ν}.

2will be applied for λ^{+}

g_{ηˆ ⟨0⟩} = g_{ηˆ ⟨1⟩}.

This is clearly possible. Let N_{η}^{0} = M^{∗} ↾ Rang(g_{η}). For η ∈ ^{λ^{+}}2 let N_{η}^{0} = M^{∗} ↾ ⋃_{α<λ^{+}} Rang(g_{η↾α}) and let g_{η} = ⋃_{α<λ^{+}} g_{η↾α}. Chasing arrows we can find for η ∈ ^{λ^{+}}2

a limit to (⟨M_{η↾α}^{0}, M_{η↾α}^{1}, a_{η↾α}⟩, h_{η↾β,η↾α} : α < β < λ^{+}) say (M_{η}^{0}, M_{η}^{1}, a_{η}) ∈ K_{λ^{+}}^{3} and h_{η,ν} for ν ◁ η as usual. Let f_{η} be the function from M_{η}^{0} into M^{∗} such that for α < λ^{+} we have f_{η} ∘ h_{η,η↾α} = g_{η}. So f_{η} is a ≤_{𝔎}-embedding of M_{η}^{0} into M^{∗}. So we can extend f_{η} to f_{η}^{+}, a ≤_{𝔎}-embedding of M_{η}^{1} into M^{∗}.

Let a_{η}^{∗} = f_{η}^{+}(a_{η}) for η ∈ ^{λ^{+}}2.

As 2^{λ^{+}} > λ^{++} for some η_{0} ≠ η_{1} we have a_{η_{0}}^{∗} = a_{η_{1}}^{∗}. So for some α < λ^{+}, η_{0} ↾ α = η_{1} ↾ α but η_{0}(α) ≠ η_{1}(α), without loss of generality η_{ℓ}(α) = ℓ and by clause (v) above we get a contradiction. □_{2.7}

2.8 Claim. 1) Assume (∗)_{λ}^{2} or just

(∗)_{λ}^{2^{−}} 𝔎 has amalgamation in λ and LS(𝔎) ≤ λ.

If M_{0} ≤_{𝔎} N_{0} ∈ K_{λ} and (M_{0}, M_{1}, a) ∈ K_{λ}^{3} \underline{then} there is N ∈ K_{≤λ^{+}} such that: N_{0} ≤_{𝔎} N and for every c ∈ N satisfying tp(c, M_{0}, N) = tp(a, M_{0}, M_{1}), there is a ≤_{𝔎}-embedding h of M_{1} into N extending id_{M_{0}} such that h(a) = c and N ∉ K_{λ^{+}} ⇒ N is a <_{𝔎}-maximal member of K_{λ}.

2) Assume M_{0} ≤_{𝔎} N_{0} ∈ K_{λ} and (M_{0}, M_{1}, a) ∈ K_{λ}^{3} and every triple in K_{λ}^{3} has the weak extension property. \underline{Then} there is N ∈ K_{λ^{+}} such that: N_{0} ≤_{𝔎} N and for every c ∈ N either for some N′ ∈ K_{λ} we have N_{0} ∪ {c} ⊆ N′ ≤_{𝔎} N and c does not strongly realize tp(a, M_{0}, M_{1}) \underline{or} there is an ≤_{𝔎}-embedding h of M_{1} into N extending id_{M_{0}} such that h(a) = c.

3) We can in parts (1), (2) have N_{0} ∈ K_{λ^{+}}.

Proof. 1) We choose by induction on α < λ^{+}, a model N_{α} ∈ K_{λ} increasing (by ≤_{𝔎}) continuous such that: for α even N_{α} ≠ N_{α+1} if N_{α} is not ≤_{𝔎}-maximal, and for α odd let β_{α} = Min{β : β = α + 1 or β ≤ α and there is c ∈ N_{β} such that there is no ≤_{𝔎}-embedding h of M_{1} into N_{α} extending id_{M_{0}} such that h(a) = c but for some N ∈ K_{λ}, N_{α} ≤_{𝔎} N and there is a ≤_{𝔎}-embedding h of M_{1} into N extending id_{M_{0}} such that h(a) = c}, and if β_{α} ≤ α then choose N exemplifying this and let N_{α+1} = N. By the definition of type we are done.

2) Same proof, note that the non ≤_{𝔎}-maximality of N_{α} (and hence N) follows by a weak extension property.

3) By using 0.19(2) repeatedly λ^{+} times. □_{2.7}

2.9 Claim. Assume (∗)_{λ}^{2} or just:

(∗)_{λ}^{2^{−}} 𝔎 has amalgamation for λ and LS(𝔎) ≤ λ.

1) Assume that above (M_{0}, M_{1}, a) ∈ K_{λ}^{3} there is no minimal member of K_{λ}^{3} \underline{then} (M_{0}, M_{1}, a) itself has the extension property.

2) If (M_{0}, M_{1}, a) ∈ K_{λ}^{3}, M_{0} ≤_{𝔎} N ∈ K and the number of c ∈ N such that tp(c, M_{0}, N) = tp(a, M_{0}, M_{1}) is > λ \underline{then} (M_{0}, M_{1}, a) has the extension property.

3) Assume above (M_{0}, M_{1}, a) ∈ K_{λ}^{3} there is no minimal member of K_{λ}^{3} \underline{then}

(∗) for some N we have: M_{0} ≤_{𝔎} N and N is as required in part (2).

2.10 Remark. 1) See ?.

2) Note that (∗)_{λ}^{2} is from 2.2 and (∗)_{λ}^{2} ⇒ (∗)_{λ}^{2^{−}} by 2.2.

Proof. 1) Follows by part (2) and (3) (part (b)).

2) Without loss of generality N has cardinality λ^{+} and also is as in 2.8. By 0.19(2) for any M′_{0} such that M_{0} ≤_{𝔎} M′_{0} ∈ K_{λ} there is N_{1}, N ≤_{𝔎} N_{1} ∈ K_{λ^{+}} and a ≤_{𝔎}-embedding h of M′_{0} into N_{1} extending id_{M_{0}}. Now some c ∈ N\h(M′_{0}) realizes tp(a, M_{0}, M_{1}) and there is an embedding h_{c} of M_{1} into N extending id_{M_{0}} such that h_{c}(a) = c. Lastly let N′_{1} ≤_{𝔎} N_{1} be of cardinality λ and include Rang(h_{c})∪ Rang(h) (send a to c via h_{c}). So modulo chasing arrows we have proved that (M_{0}, M_{1}, a) has the extension property for the case M′_{0} ∈ K_{λ}, M_{0} ≤_{𝔎} M′_{0}, which was arbitrary so we are done.

3) We first prove

(∗)_{0} For some M_{0}^{+}, M_{0} ≤_{𝔎} M_{0}^{+} ∈ K_{λ} and tp(a, M_{0}, M_{1}) has > λ extensions in 𝒮 (M_{0}^{+}) (in fact ≥ min{2^{μ} : 2^{μ} > λ}).

Proof of (∗)_{0}. Let M_{η}^{0}, M_{η}^{1}, a_{η}, h_{η,ν} (η ∈ ^{λ^{+}>}2 and ν ⊴ η) be as in the proof of 2.7 (i.e. satisfy (i) − (vi) there) and M_{<>}^{0} = M^{0}, M_{<>}^{1} = M^{1}. Let μ = Min{μ : 2^{μ} > λ}, so ^{μ>}2 has cardinality ≤ λ and μ ≤ λ. Let ^{μ>}2 = {η_{ζ} : ζ < ζ^{∗}} be such that η_{ξ} ◁ η_{ζ} ⇒ ξ < ζ and so ζ^{∗} < λ^{+} and without loss of generality is a limit ordinal. Now we can choose by induction on ζ ≤ ζ^{∗} a model M_{ζ}^{∗} ∈ K_{λ} and, if ζ < ζ^{∗} also a function g_{η_{ζ}} such that:

(α) M_{ζ}^{∗} is ≤_{𝔎}-increasing continuous in ζ

(β) M_{0}^{∗} = M_{⟨⟩}

(γ) g_{η_{ζ}} is a ≤_{𝔎}-embedding of M_{η_{ζ}}^{0} into M_{ζ+1}^{∗}

(δ) if η_{ξ} ◁ η_{ζ} then g_{η_{ζ}} ∘ h_{η_{ζ},η_{ξ}} = g_{η_{ξ}}

(ε) if ξ < ξ_{0}, ξ < ξ_{1}, η_{ξ_{0}} = η_{ξ}ˆ ⟨0⟩, η_{ξ_{1}} = η_{ξ}ˆ ⟨1⟩ then (M_{ξ_{0}}^{∗} = M_{ξ_{1}}^{∗} and) g_{η_{ξ_{0}}} = g_{η_{ξ_{1}}} .

So for η ∈ ^{μ}2 we can find g_{η}, a ≤_{𝔎}-embedding of M_{η}^{0} into M_{ζ^{∗}}^{∗}, such that g_{η} ∘ h_{η,η↾α} = g_{η↾α} for every α < μ. We also can let p_{η}^{0} = g_{η}[tp(a_{η}, M_{η}^{0}, M_{η}^{1})] ∈ 𝒮 (M_{ζ^{∗}}^{∗} ↾ Rang(g_{η})), and find p_{η} such that p_{η}^{0} ≤ p_{η} ∈ 𝒮 (M_{ζ^{∗}}^{∗}) (possible as 𝔎_{λ} has amalgamation by 2.2(1) if (∗)_{λ}^{2} holds and by (∗)_{λ}^{2^{−}} otherwise).

For η ∈ ^{μ}2 and α ≤ μ let N_{η↾α}^{0} = M_{ζ^{∗}}^{∗} ↾ Rang (g_{η↾α}). Clearly for η ∈ ^{μ≥}2, N_{η}^{0} is well defined; η ◁ ν ∈ ^{μ≥}2 ⇒ N_{η}^{0} ≤_{𝔎} N_{ν}^{0}; and N_{ηˆ ⟨0⟩}^{0} = N_{ηˆ ⟨1⟩}^{0}. Also letting for η ∈ ^{μ}2, and α ≤ μ, the type p_{η↾α}^{0} be p_{η}^{0} ↾ N_{η↾α}^{0} we have: p_{η}^{0} ∈ 𝒮 (N_{η}^{0}) is well defined, η ◁ ν ∈ ^{μ≥}2 ⇒ p_{η}^{0} ≤ p_{ν}^{0} and p_{ηˆ ⟨0⟩}^{0} ≠ p_{ηˆ ⟨1⟩}^{0}. Hence for η_{0} ≠ η_{1} from ^{μ}2 we have p_{η_{0}} ≠ p_{η_{1}}. So |𝒮 (M_{ζ^{∗}}^{∗})| ≥ 2^{μ} > λ.

Proof of (∗). We choose by induction on i < λ^{+}, N_{i} ∈ K_{λ} which is ≤_{𝔎}-increasing continuous, N_{0} = M_{0}^{+} (M_{0}^{+} is from (∗)_{0} above) and for each i some c_{i} ∈ N_{i+1} realizes over N_{0} = M_{0}^{+} a (complete) extension of p = tp(a, M_{0}, M_{1}) not realized in N_{i}. There is such type by clause (∗)_{0} above and there is such an N_{i+1} as 𝔎 has amalgamation in λ. Clearly c_{i} ∉ N_{i} and so ⋃_{i<λ^{+}} N_{i} is as required. □_{2.9}

2.11 Claim. Assume (∗)_{λ^{+}}^{2^{−}} (from 2.9; that is 𝔎 has amalgamation in λ and LS(𝔎) ≤ λ).

If (M_{0}, M_{1}, a) ≤ (M′_{0}, M′_{1}, a) are from K_{λ}^{3}, and the second has the extension property, \underline{then} so does the first.

Proof. Use amalgamation over M_{0}: if M_{0} ≤_{𝔎} N_{0} ∈ K_{λ} we can find N′_{0} such that M′_{0} ≤_{𝔎} N′_{0} ∈ K_{λ} and there is a ≤_{𝔎}-embedding of N_{0} into N′_{0} over M_{0}. Now use “(M′_{0}, M′_{1}, a) has the extension property” for N′_{0}. □_{2.11}

Now we introduce

2.12 Definition. For any models M, M_{0} ∈ K_{λ}, any type p ∈ 𝒮 (M_{0}) and f_{0} : M_{0} ⟶_{iso}^{onto} M we let 𝒮_{p}(M) = 𝒮_{M}^{p} = {f_{0}(f(p)) : f ∈ AUT(M_{0})}. Note: 𝒮_{M}^{p}

does not depend on f_{0}. If K is categorical in λ, 𝒮_{p}(M) is well defined for every M ∈ K_{λ}. We write also 𝒮_{tp(a,M_{0},M_{1})}(M) or 𝒮_{(M_{0},M_{1},a)}(M) when (M_{0}, M_{1}, a) ∈ K_{λ}^{3}.

2.13 Claim. Assume (∗)_{λ}^{2^{−}} + (∗)_{λ^{+}}^{2} + (∗)_{λ}^{3} + 2^{λ^{+}} < 2^{λ^{++}} + K_{λ^{+3}} = ∅. If (M_{0}, M_{1}, a) ∈ K_{λ}^{3} is minimal \underline{then} it has the extension property.

Remark. Instead of 2^{λ^{+}} < 2^{λ^{++}}, we can just demand the definable weak diamond.

Proof. Assume not. By the previous two claims (2.9(1), 2.11) we may assume that (M_{0}, M_{1}, a) is minimal. As 𝔎 has amalgamation in λ^{+} by 2.2(1), there is M^{∗} ∈ K_{λ^{++}} which is saturated above λ^{+} (as K_{λ^{+3}} = ∅) hence M^{∗} is saturated above λ (by 1.10(3)). By 2.6(1) + 2.11, without loss of generality

⨂0 (M_{0}, M_{1}, a) is reduced.

Let h : M_{0} → M^{∗} be a ≤_{𝔎}-embedding and let p = tp(a, M_{0}, M_{1}). If h(p) is realized in M^{∗} by ≥ λ^{+} elements we are done by 2.9(2). So assume

⨂1 h(p) is realized by ≤ λ members of M^{∗}.

Similarly

⨂_{1}^{+} q is realized by ≤ λ members of M^{∗} for q = g(tp(a, M′_{0}, M′_{1})) if g is a ≤_{𝔎}-embedding of M′_{0} into M^{∗} and (M′_{0}, M′_{1}, a) ≥ (M_{0}, M_{1}, a).

Next we prove

⨂2 for some reduced (M′_{0}, M′_{1}, a) ≥ (M_{0}, M_{1}, a) from K_{λ}^{3} we have |𝒮_{(M′_{0},M′_{1},a)}(M′_{0})| > λ^{+}.

Proof of ⨂_{2}. If not, we build two non-isomorphic members of K_{λ^{+}} as follows.

\underline{First}: Choose by induction on i < λ^{+}, (N_{0,i}, N_{1,i}, a) ∈ K_{λ}^{3} reduced (see 2.6(1)), increasing continuously (see 2.6(2)), with N_{0,i} ≠ N_{0,i+1}, (N_{0,0}, N_{1,0}, a) = (M_{0}, M_{1}, a); this is possible as (N_{0,i}, N_{1,i}, a) ∈ K_{λ}^{3} has the weak extension property (by 2.4 see

2.3(1)). Let N^{1} = ⋃_{i<λ^{+}} N_{0,i}.

\underline{Second}: Choose by induction on i < λ^{+}, N_{i}^{0} ≤_{𝔎} M^{∗}, ∥N_{i}^{0}∥ = λ, N_{i}^{0} strictly increasing continuous such that:

(∗) for every β < λ^{+}, i < λ^{+}, and q ∈ 𝒮_{(N_{0,i},N_{1,i},a)}(N_{β}^{0}) for some γ ∈ (β, λ^{+}) there are no N′, N_{γ}^{0} ≤_{𝔎} N′ ∈ K_{λ} and c ∈ N′\N_{γ}^{0} such that c realize q.

This is straightforward by ⊗_{1} + ⊗_{2} and bookkeeping. Let N^{0} = ⋃_{i<λ^{+}} N_{i}^{0}. By

categoricity of K_{λ^{+}} there is an isomorphism g from N^{1} onto N^{0}, so E = {δ < λ^{+} : g maps N_{0,δ} onto N_{δ}^{0}} is a club of λ^{+}. Now let δ^{∗} ∈ E, and apply (∗) for β = δ^{∗}, q = g(tp(a, N_{0,δ^{∗}}, N_{1,δ^{∗}})) to get γ. Choose δ ∈ E which is > γ. Now N_{1,δ} gives a contradiction. □_{⊗_{2}}

Without loss of generality

⨂3 |𝒮_{(M_{0},M_{1},a)}(M_{0})| > λ^{+} and p = tp(a, M_{0}, M_{1}).

Next we claim

⨂4 If M ∈ K_{λ}, M ≤_{𝔎} M^{∗}, Γ ⊆ ⋃{𝒮_{p}(M′) : M′ ≤_{𝔎} M, ∥M′∥ = λ}, |Γ| ≤ λ^{+}, then

Γ^{∗} =: {q ∈ 𝒮_{p}(M) : there is M′, M ≤_{𝔎} M′, ∥M′∥ = λ,

M′ realizes q but no r ∈ Γ}

has cardinality λ^{++}, in fact |𝒮_{p}(M)\Γ^{∗}| ≤ λ^{+}.

Proof of ⊗_{4}. Without loss of generality |M^{∗}| = λ^{++} (i.e. the universe of M^{∗} is λ^{++}). For every q ∈ 𝒮_{p}(M) there is a triple (M_{0}, M_{1,q}, a_{q}) isomorphic to (M_{0}, M_{1}, a) (hence reduced) such that tp(a_{q}, M_{0}, M_{1,q}) = q. As M^{∗} is saturated above λ, by 0.25 without loss of generality M_{1,q} ≤_{𝔎} M^{∗}.

Without loss of generality δ < λ^{++} & (λ^{+} divides δ) ⇒ M_{δ} =: M^{∗} ↾ δ ≤_{𝔎} M^{∗}. Now

(∗)_{0} q_{1} ≠ q_{2} ⇒ a_{q_{1}} ≠ q_{q_{2}} and

(∗)_{1} a_{q} ∉ δ & δ < λ^{++} & λ^{+} divides δ ⇒ M_{1,q} ∩ M_{δ} = M.

[Why? As (M, M_{1,q}, a_{q}) is reduced]. Now if r ∈ Γ, say r ∈ 𝒮_{p}(M′′) then by ⊗_{1} we know A_{r} = {c ∈ M^{∗} : c realizes r} has cardinality ≤ λ and hence A = ⋃{A_{r} : r ∈ Γ} has cardinality ≤ λ^{+}, so we can find δ < λ^{++} divisible by λ^{+} such that A ⊆ δ. But (by ⊗_{3}) we have |𝒮_{p}(M)| > λ^{+} hence we can find q[δ] ∈ 𝒮_{p}(M) such that a_{q(δ)} ∉ δ hence (M, M_{1,q[δ]}, a_{q[δ]}), exemplifies the conclusion of ⊗_{4}. □_{⊗_{4}}

\underline{Final contradiction}: By ⊗_{4} we can construct 2^{λ^{+}} non-isomorphic members of K_{λ^{+}} using 1.6(1) as follows. We choose by induction on α < λ^{+}, for every η ∈ α2, M_{η}, p_{η}^{0}, p_{η}^{1} such that:

(a) M_{<>} = M_{0}

(b) M_{η} ∈ K_{λ}

(c) ⟨M_{η↾β} : β ≤ ℓg(η)⟩ is <_{𝔎}-increasing continuous

(d) p_{η} ∈ 𝒮_{p}(M_{η↾β})

(e) for β ≤ α, we have p_{η}^{0}, p_{η}^{1} ∈ 𝒮 (M_{η}) and: M_{η} realizes p_{η↾β}^{ℓ} iff β < α & ℓ = η(β).

If α = 0 or α is a limit, there is no problem to define M_{η} for η ∈ ^{α}2. If M_{η} is defined, we can choose by induction on i < λ^{++}, (N_{η,i}, a_{η,i}) such that (M_{η}, N_{η,i}, a_{η,i}) ∈ K_{λ}^{3}, tp(a_{η,i}, M_{η}, N_{η,i}) ∈ 𝒮_{p}(M_{η}) and N_{η,i} omits any q ∈ {p_{η↾β}^{ℓ} : β < ℓg(η), ℓ ≠ η(β)} ∪ {tp(b, M_{η}, N_{η,j}) : j < i and b ∈ N_{η,j} and tp(b, M_{η}, N_{η,j}) ∈ 𝒮_{p}(M_{η})}. By ⊗_{4} we can choose (N_{η,j}, a_{η,j}). Hence |W_{η,i}| ≤ λ where W_{η,i} = {j < λ^{++} : for some b ∈ N_{η,i} we have tp(b, M_{η}, N_{η,i}) = tp(a_{η,j}, M_{η}, N_{η,j})}. Hence we can find i < j < λ^{++} such that i ∉ W_{η,j} & j ∉ W_{η,i}. Let M_{ηˆ ⟨0⟩} = N_{η,i}, M_{ηˆ ⟨1⟩} = N_{η,j}, p_{η}^{0} = tp(a_{η,i}, M_{η}, N_{η,i}), p_{η}^{1} = tp(a_{η,j}, M_{η}, N_{η,j}). Let for η ∈ ^{λ^{+}}2, M_{η} = ⋃_{α<λ^{+}} M_{η↾α}, and apply 1.6(1). □_{2.12}

2.14 Conclusion. [(∗)_{λ}^{2^{−}} + (∗)_{λ^{+}}^{2} + (∗)_{λ}^{3} + 2^{λ^{+}} < 2^{λ^{++}} + K_{λ^{+3}} = ∅].

Every (M_{0}, M_{1}, a) ∈ K_{λ}^{3} has the extension property.

Proof. By 2.11 and 2.9 + 2.12. □_{2.14}

2.15 Remark. Conclusion 2.14 says in other words: \underline{if}

(a) LS(𝔎) ≤ λ

(b) K is categorical in λ and in λ^{+}

(c) 1 ≤ I(λ^{++}, K) < 2^{λ^{++}}

(d) K_{λ^{+3}} is empty

(e) 2^{λ^{+}} < 2^{λ^{++}} (or just definable weak diamond)

(f) 𝔎 has amalgamation in λ

\underline{then} every triple (M_{0}, M_{1}, a) in K_{λ}^{3} has the extension property.

2.16 Claim. [(∗)_{λ}^{3}, in other words LS(𝔎) ≤ λ; 𝔎 categorical in λ and in λ^{+}; and 1 ≤ I(λ^{++}, K) < 2^{λ^{++}}].

If M_{0} ≤_{𝔎} M_{1} are in K_{λ} \underline{then} we can find α < λ^{+} and ⟨N_{i} : i ≤ α⟩ which is ≤_{𝔎}-increasing continuous, N_{i} ∈ K_{λ}, (N_{i}, N_{i+1}, a_{i}) ∈ K_{λ}^{3} is reduced, M_{0} = N_{0}, and M_{1} ≤_{𝔎} N_{α}.

Proof. If not, we can contradict categoricity in K_{λ^{+}} (similar to the proof of ⊗_{2} during the proof of 2.12).

Without loss of generality M_{0} ≠ M_{1}. We choose by induction on i < λ^{+}, N_{i}^{0} ∈ K_{λ}, ≤_{𝔎}-increasing continuous such that (N_{i}^{0}, N_{i+1}^{0}) ≅ (M_{0}, M_{1}) (possible by 2.6(9) and the categoricity of K in λ). Let N^{0} = ⋃_{i<λ^{+}} N_{i}^{0}.

We choose by induction on i < λ^{+}, N_{i}^{1} ∈ K_{λ}, ≤_{𝔎}-increasing continuous and a_{i} such that (N_{i}^{1}, N_{i+1}^{1}, a_{i}) ∈ K_{λ}^{3} is reduced and let N^{1} = ⋃_{i<λ^{+}} N_{i}^{1} (possible by

2.6(1) and the categoricity of K in λ). So by the categoricity in λ^{+} without loss of generality N^{1} = N^{0}, hence for some δ_{1} < δ_{2} < λ^{+} we have

N_{δ_{1}}^{0} = N_{δ_{1}}^{1}, N_{δ_{2}}^{0} = N_{δ_{2}}^{1}.

By changing names (N_{δ_{1}}^{0}, N_{δ_{1}+1}^{0}) = (M_{0}, M_{1}) and so ⟨N_{δ_{2}+i} : i ≤ δ_{2} − δ_{1}⟩ is as required. □_{2.16}

2.17 Conclusion. [(∗)_{λ}^{2^{−}} + (∗)_{λ^{+}}^{2} + (∗)_{λ}^{3} + 2^{λ^{+}} < 2^{λ^{++}} + K_{λ^{+3}} = ∅, i.e. the assumption of 2.14]. K_{λ} has disjoint amalgamation (M_{2}, M_{1} are in disjoint amalgamation over M_{0} in M_{3} if M_{0} ≤_{𝔎} M_{ℓ} ≤_{𝔎} M_{3}, M_{1} ∩ M_{2} = M_{0}).

Proof. By 2.16 and iterated applications of 2.14. □_{2.17}

§3 Non-structure

The first major aim of this section is to prove the density of minimal types using as set theoretic assumptions only 2^{λ} < 2^{λ^{+}} < 2^{λ^{++}} from cardinal arithmetic. The second aim is to prepare for a proof of a weak form of uniqueness of amalgamation in 𝔎_{λ}. Our aim is also to explain various methods. The proofs are similar to the ones in [Sh 87b, §6].

The immediate role of this section is to get many models in λ^{++} from the assumption “the minimal triples in K_{λ}^{3} are not dense”: in ? we get this under some additional assumptions, and in ? we get it using only the additional assumption I(λ, K^{+3}) = 0, which suffices for our main theorem (this does not suffice for the theorem of [Sh 600], hence is eliminated there).

But the section is done in a more general fashion, so let us first explain two general results concerning the construction of many models based on repeated “failures of amalgamation” or “nonminimality of types”.

In 3.19, we give a construction assuming the ideal of small subsets of λ^{+} (that is WDmId(λ^{+})) is not λ^{++}-saturated, as exemplified by ⟨S_{α} : α < λ^{++}⟩. We build

for η ∈ ^{(λ^{+2})>}2 models M_{η} ∈ K_{λ^{+}} such that M_{η} = ⋃_{α<λ^{+}} M_{η,α}, |M_{η}| = λ × (1 + ℓgη)

and ν ◁ η ⇒ M_{ν} ≤_{𝔎} M_{η}. Building M_{ηˆ ⟨ℓ⟩} manufacture M_{ηˆ ⟨ℓ⟩} as a limit of models ⟨M_{ηˆ ⟨ℓ⟩,α} : α < λ^{+}⟩, a representation of M_{ηˆ ⟨ℓ⟩}, usually in a way predetermined simply, except when α ∈ S_{ℓg(η)} and ℓ = 1, and then we consult a weak diamond sequence. This is like 1.6(1), but there we use our understanding of models in K_{λ} to build many models in K_{λ^{+}} while here we build models in K_{λ^{++}}, thus getting 2^{λ^{+2}} models in λ^{+2}. We even get 2^{(λ^{+2})} models in K_{λ^{+2}} with none ≤_{𝔎}-embeddable into any other.

A second proof 3.23 is like 1.4(1) in the sense that we get only close to 2^{λ^{++}} models. It is similar to [Sh 87b, 6.4], and the parallel to [Sh 87b, 6.3] holds here. So we have to find an analog of [Sh 87b, definition 6.5,6.7]. But there we use fullness on the side (meaning: M ∈ K_{λ} is full over N ∈ K_{λ} if N ≤_{𝔎} M, and (M, c)_{c∈N} is saturated), but we do not have this yet.

We still have not explained the framework of this section. In 3.1 - 3.5 we present construction frameworks C, which involve sequences of models of length ≤ λ each of cardinality < λ and in particular, define local and nice C. In our applications here, λ^{+} plays the role of λ (and < λ^{+} is specialized to λ). Then in 3.6 - 3.8 we present examples of such frameworks. Our intention is to use the limit of a sequence ⟨M_{α} : α < λ⟩ as an approximation to a model of cardinality λ^{+}. For this we define in 3.10 - 3.11 a successor relation (next approximation), modulo a “< λ-amalgamation choice function”; this is denoted M̄^{1} ≤_{F_{1}}^{at} M̄^{2}. Iterating it we get the quasi order ≤_{F} (see 3.13). In 3.14 we define the key coding properties, (of an amalgamation choice function F for the framework C). The intention is that these coding properties suffice to build many non-isomorphic models in λ^{+}. In 3.17 we give the “atomic step” for this construction.

In 3.18 we prove the existence of 2^{λ^{+}} non-isomorphic models, using the λ-coding property. As we do not have this in some applications we have in mind, we next turn to the weak λ-coding property in 3.19 as well as the weak (local) λ-coding property and corresponding properties of F (all in Definition 3.20, 3.22), connect them 3.21, and prove that there are many models in 3.23. Lastly, ?, ? deal with our concrete case: if the minimal triples in 𝔎_{λ}^{3} are not dense then in most cases failures of amalgamation lead to the λ^{+}-coding property and hence to many models in cardinality λ^{++}.

Note generally that we mainly axiomatize the construction of models in λ^{+}, not how we get M̄′, M̄ ≤_{F,a}^{at} M̄′ ∈ Seq_{λ} that is coding properties; for the last point, see the examples just cited.

Later, in 6.10, we shall need again to use the machinery from this section, in trying to prove that there are enough cases of disjoint amalgamation in 𝔎_{λ}. We may want to turn the framework presented here into a more general one; see more in [Sh 600].

3.1 Context.

1) 𝔎 is an abstract elementary class.

2) But =_{M} or =_{𝔎} is just an equivalence relation, i.e. for M ∈ K, =^{M} is an equivalence relation on |M|, moreover a congruence relation relative to all relations and functions in τ(M) that is for R ∈ τ(M) an n-ary relation, we have

⋀_{ℓ<n} a_{i} =^{M} b_{i} ⇒ ⟨a_{0}, …, a_{n−1}⟩ ∈ R^{M} ≡ ⟨b_{0}, …, b_{n−1}⟩ ∈ R^{M}.

We let ∥M∥ = |(M)/ =^{M} | and

K_{λ,μ} = {M : |M| has μ elements and |M|/ =^{M} has λ elements}.

3) Now the meaning of ≤_{𝔎} should be clear but M <_{𝔎} N means (M ∈ K, N ∈ K and) M ⊆ N and M/ =^{M}≤_{𝔎} N/ =^{N} and ∃a ∈ N[a/ =^{N} ∉ (M/ =^{N})] i.e. (∃a ∈ N)(∀b ∈ M)(¬a =^{N} b).

4) K_{λ}^{3} = {(M, N, a) : M ≤_{𝔎} N are from K_{λ,λ} and a ∈ N, (a/ =^{N}) ∉ (M/ =^{N})}.

5) In this context “R is an isomorphism relation from M_{1} onto M_{2}” if

(a) R ⊆ M_{1} × M_{2}

(b) a_{1} =^{M_{1}} b_{1} & a_{2} =^{M_{2}} b_{2} ⇒ a_{1}Ra_{2} ↔ b_{1}Rb_{2}

(c) (∀x ∈ M_{1})(∃y ∈ M_{2})xRy

(d) (∀y ∈ M_{2})(∃x ∈ M_{1})xRy

(e) if Q ∈ τ(M_{1}) = τ(M_{2}) is an n-place relation and a_{ℓ}Rb_{ℓ} for ℓ = 0, …, n − 1 then (a_{0}, …, a_{n−1}) ∈ Q^{M_{1}} → (b_{0}, …, b_{n−1}) ∈ Q^{M_{2}}

(f) a_{1}Ra_{2} & b_{1}Rb_{2} ⇒ (a_{1} =^{M_{1}} b_{1} ⇔ a_{2} =^{M_{2}} b_{2}).

3.2 Explanation. The need of 3.1(2) is just to deal with amalgamations which are not necessarily disjoint. If we use disjoint amalgamation, we can omit 3.1(2) below on Definition 3.10, a disappears so F is four place and use K_{λ} instead of K_{λ,λ}. This is continued in [Sh 600, 2.17]. May be 3.1 would be better understood after reading 3.10, after clause (c).

3.3 Definition. Let λ be regular uncountable and 𝔎 an abstract elementary class.

1) A λ-construction framework C = C(𝔎^{+}, Seq, ≤^{∗}) means (we shall use it below with λ^{+} playing the role of λ):

(a) τ^{+} = τ^{+}(𝔎^{+}) is a vocabulary extending τ. 𝔎^{+} is an abstract elementary class satisfying axioms I,II,III from 0.5 and M ≤_{𝔎^{+}} N ⇒ M ↾ τ ≤_{𝔎} N ↾ τ. Furthermore 𝔎^{+} = 𝔎_{<λ}^{+}. As above, equality (in τ) is just a congruence relation.

(b) Seq = ⋃_{α≤λ} Seq_{α} where for α ≤ λ, Seq_{α} is a subset of

{M̄ : M̄ = ⟨M_{i} : i < α⟩, M_{i} ∈ 𝔎^{+} is ≤_{𝔎^{+}}-increasing continuous}. For α = λ we require further that M/ =^{M} has cardinality λ, where

M = ⋃_{i<λ} M_{i}.

(c) ≤^{∗} is a three place relation on triples x, y, t written x ≤_{t}^{∗} y for x, y ∈ Seq and t a set of pairwise disjoint closed intervals of ℓg(x).

We require:

(d) Seq is closed under isomorphism and initial segments.

(e) If M̄^{1} ≤_{t}^{∗} M̄^{2} and γ ∈ ∪t \underline{then} M_{γ}^{1} ≤_{𝔎^{+}} M_{γ}^{2} and hence M_{γ}^{1} ↾ τ ≤_{𝔎} M_{γ}^{2} ↾ τ and γ < ℓg(M̄^{2}).

(f) If M̄^{1} ≤_{t}^{∗} M̄^{2}, s ⊆ t, and M̄^{2} ⊴ M̄^{3} ∈ Seq \underline{then} M̄^{1} ≤_{s}^{∗} M̄^{3}.

(g) If t is a set of closed pairwise disjoint intervals of ℓg(M̄) and M̄ ∈ Seq \underline{then} M̄ ≤_{t}^{∗} M̄.

\underline{3.4 Convention/Definition}:

1) From now on C will be a λ-construction framework.

2) If M̄ ∈ Seq_{λ} \underline{then} we let M̄ = ⟨M_{i} : i < λ⟩ and M =: ⋃_{i<λ} M_{i}; similarly with

M̄^{x} = ⟨M_{i}^{x} : i < λ⟩.

3) K_{λ}^{qr} = {(M̄, f) : M̄ ∈ Seq_{λ} and f : λ → λ}.

4) If (M̄^{ℓ}, f^{ℓ}) ∈ K_{λ}^{qr} for ℓ = 1, 2 \underline{then} (M̄^{1}, f^{1}) ≤ (M̄^{2}, f^{2}) means that: for some club E of λ, we have

(a) δ ∈ E ⇒ f^{1}(δ) ≤ f^{2}(δ) and

(b) M̄^{1} ≤_{t}^{∗} M̄^{2} where t = t_{E,f^{1}} = {[δ, δ + f^{1}(δ)] : δ ∈ E}.

5) Seq^{s} = {M̄ ∈ Seq : ⋃_{i} |M_{i}| is a set of ordinals < λ^{+}}; similarly for Seq_{α}.

6) K_{λ}^{qs} = {(M̄, f) ∈ K_{λ}^{qr} : M̄ ∈ Seq_{λ}^{s}}.

7) C is \underline{local} (respectively, revised local) if the following clauses (a), (b), (c) hold:

(a) M̄ = ⟨M_{i} : i < α⟩ ∈ Seq_{α} iff:

(α) M̄ is ≤_{𝔎^{+}}-increasing continuous in 𝔎_{<λ}^{+} (β) i + 1 < α ⇒ ⟨M_{i}, M_{i+1}⟩ ∈ Seq_{2}

(γ) if α = λ then |M/ =^{M} | = λ, (recall M = ⋃_{i<λ} M_{i})

(b) for M̄^{1}, M̄^{2} ∈ Seq and t a set of pairwise disjoint closed intervals contained in ℓg(M̄^{1}) we have:

M̄^{1} ≤_{t}^{∗} M̄^{2} \underline{iff} [γ_{1}, γ_{2}] ∈ t implies

(α) γ ∈ [γ_{1}, γ_{2}] ⇒ M_{γ}^{1} ≤_{𝔎^{+}} M̄_{γ}^{2}

(β) in the local case: γ ∈ [γ_{1}, γ_{2}) ⇒ ⟨M_{γ}^{1}, M_{γ+1}^{1}⟩ ≤_{{[0,1]}}^{∗} ⟨M_{γ}^{2}, M_{γ+1}^{2}⟩; in the revised local case: if ℓg(M̄^{1}), ℓg(M̄^{2}) < λ \underline{then} γ ∈ [γ_{1}, γ_{2}] ⇒ ⟨M_{γ}^{1}, M^{1}⟩ ≤_{{[0,1]}}^{∗} ⟨M_{γ}^{2}, M^{2}⟩, and generally for some

club E of λ, γ ∈ [γ_{1}, γ_{2}] & γ < δ ∈ E ⇒ ⟨M_{γ}^{1}, ⋃_{β<δ} M_{β}^{1}⟩ ≤_{{[0,1]}}^{∗}

⟨M_{γ}^{2}, ⋃_{β<δ} M_{β}^{2}⟩ (and if ℓg(M̄^{ℓ}) = α_{ℓ} < δ \underline{then} ⋃_{β<δ} M_{β}^{ℓ} means ⋃_{β<α^{ℓ}} M_{β}^{ℓ}).

(c) if ⟨M_{0}^{ζ}, M_{1}^{ζ}⟩ ∈ Seq_{2} for ζ < ζ^{∗} < λ, ⟨M_{0}^{ζ} : ζ ≤ ζ^{∗}⟩ and ⟨M_{1}^{ζ} : ζ ≤ ζ^{∗}⟩ are ≤_{𝔎^{+}}-increasing continuous, and ζ < ζ^{∗} ⇒ ⟨M_{0}^{ζ}, M_{1}^{ζ}⟩ ≤_{{[0,1]}}^{∗} ⟨M_{0}^{ζ+1}, M_{1}^{ζ+1}⟩

\underline{then} ⟨M_{0}^{0}, M_{1}^{0}⟩ ≤_{\{[0,1]\}}^{∗} ⟨M_{0}^{ζ^{∗}}, M_{1}^{ζ^{∗}}⟩ ∈ Seq_{2}.

So intervals [α, α] ∈ t are essentially irrelevant for the local version they just require M_{α}^{1} ≤_{𝔎^{+}} M_{α}^{2}. In the revised local version it is natural to add monotonicity for ≤_{{[0,1]}}.

8) For α ≤ λ we say C is \underline{closed} for α if:

(α) M̄ = ⟨M_{i} : i < α⟩ ∈ Seq \underline{iff} β < α ⇒ M̄ ↾ (β + 1) ∈ Seq (β) if M̄^{ℓ} = ⟨M_{i}^{ℓ} : i < α_{ℓ}⟩ ∈ Seq for ℓ = 1, 2 and α = α_{1} < α_{2}, \underline{then}

M̄^{1} ≤_{t}^{∗} M̄^{2} ⇔ M̄^{1} ≤_{t}^{∗} M̄^{2} ↾ α.

9) C is \underline{disjoint} if: M̄^{1} ≤_{t}^{∗} M̄^{2}, [γ_{1}, γ_{2}] ∈ t, γ ∈ [γ_{1}, γ_{2}) implies M_{γ}^{1} = M_{γ+1}^{1} ∩ M_{γ}^{2}. C is \underline{truly disjoint} if: M̄^{1} ≤_{t}^{∗} M̄^{2}, [γ_{1}, γ_{2}] ∈ t, γ ∈ [γ_{1}, γ_{2}] implies M_{γ}^{1} = M^{1} ∩ M_{γ}^{2}.

10) In K_{λ}^{qr}, we say (M̄, f) is a \underline{m.u.b.} (minimal upper bound) of ⟨(M̄^{ξ}, f^{ξ}) : ξ < δ⟩ if

(a) ξ < δ ⇒ (M̄^{ξ}, f^{ξ}) ≤ (M̄, f) and

(b) for any (M̄, f′) satisfying (a), for some club E of λ we have: if α ∈ E and j ≤ f(α) \underline{then} f(α) ≤ f′(α) and M_{α+j} ≤_{𝔎^{+}} M′_{α+j}.

When we require an increasing sequence in K_{λ}^{qr} to be continuous we mean that a m.u.b. is used at limits.

11) We say C is \underline{explicitly local} if it is local and

(d) if ζ^{∗} < λ is a limit ordinal, ⟨M_{0}^{ζ}, M_{1}^{ζ}⟩ ∈ Seq_{2} for ζ ≤ ζ^{∗} and for ℓ = 0, 1 the sequence ⟨M_{ℓ}^{ζ} : ζ < ζ^{∗}⟩ is ≤_{𝔎}-increasing continuous, M_{ℓ}^{ζ} ≤_{𝔎^{+}} M_{ℓ}^{ζ^{∗}},

and ζ < ξ ≤ ζ^{∗} ⇒ ⟨M_{0}^{ζ}, M_{1}^{ζ}⟩ ≤_{\{[0,1]\}}^{∗} ⟨M_{0}^{ξ}, M_{1}^{ξ}⟩ \underline{then} ⟨ ⋃_{ζ<ζ^{∗}} M_{0}^{ζ}, ⋃_{ζ<ζ^{∗}} M_{1}^{ζ}⟩ is

≤_{{[0,1]}} (M_{0}^{ζ^{∗}}, M_{1}^{ζ^{∗}}).

12) C is \underline{closed} if it is closed for every ordinal ≤ λ.

13) C is semi (respectively almost) \underline{closed}, as witnessed by G, if:

(a) C is closed for every limit ordinal δ < λ;

(b) G is a function from Seq_{<λ} to Seq_{<λ} such that M̄ ◁ G(M̄);

(c) M̄ = ⟨M_{α} : α < λ⟩ belongs to Seq_{λ} if M̄ obeys G, which means: β < λ ⇒ M̄ ↾ β ∈ Seq_{β} and {α < λ : G(M̄ ↾ α) ◁ M̄} is unbounded in λ

(d) in the almost closed version, we add: G(M̄) depends on ⋃ M̄ only.

14) C is \underline{λ-nice} if

(a) ≤ is a transitive on K_{λ}^{qr};

(b) any increasing continuous sequence in K_{λ}^{qr} of length < λ^{+} has a m.u.b.; (see part (10)) (not necessarily unique)

(c) C is closed (see part (12)).

15) C is almost λ-nice (as witnessed by G) is defined similarly, replacing “closed” by “almost closed” (witnessed by G).

3.5 Claim. Let C be a local (or revised local) λ-construction framework.

1) If (M̄^{ℓ}, f^{ℓ}) ∈ K_{λ}^{qr} for ℓ = 1, 2 and E is a club of λ^{+} and (∗) below \underline{then} (M̄^{1}, f^{1}) ≤ (M̄^{2}, f^{2}) when

(∗) if δ ∈ E then M_{δ+i}^{1} = M_{δ+i}^{2} for i ≤ f^{1}(δ) and f^{1}(δ) ≤ f^{2}(δ); in the “revised local” version assume in addition that M^{1} = M^{2}.

2) ≤ is transitive and a reflexive relation on K_{λ}^{qr}.

3) Any increasing continuous sequence of pairs from K_{λ}^{qr} of length < λ^{+} has a minimal upper bound.

4) If in addition C is explicitly local (see Definition 3.4(11)) \underline{then} any increasing sequence in K_{λ}^{qr} of length < λ^{+} has a lub.

5) C is λ-nice (hence in particular, λ-closed).

Proof. 1) Check clause (b) of Definition 3.4(7) and Definition 3.4(a).

2) Use clauses (b), (c) of Definition 3.4(7)). In (c) take ζ^{∗} = 2

3) Without loss of generality the elements of the sequence are (M̄^{ξ}, f^{ξ}) ∈ K_{λ}^{qr} for ξ < μ, where μ is a regular cardinal ≤ λ. For ξ < ζ < μ, let E_{ξ,ζ} be a closed unbounded subset of λ exemplifying Definition 3.4(4) for (M̄^{ξ}, f^{ξ}) ≤ (M̄^{ζ}, f^{ζ}). First when μ < λ, let E ⊆ ⋂_{ξ<ζ<μ} E_{ξ,ζ} ⊂ λ be a closed unbounded subset of λ, such

that: α ∈ E ⇒ α + (sup_{ξ<μ}f^{ξ}(α)) + 1 < Min(E\(α + 1)). Second when μ = λ let

E ⊆ {δ < λ : δ ∈ ⋂_{ξ<ζ<δ} E_{ξ,ζ}} ⊆ λ be a closed unbounded subset of λ such that

α ∈ E ⇒ α + sup_{ξ<α}f^{ξ}(α) + 1 < Min(E\(α + 1)). We concentrate on the first case for

notational simplicity. Let E = {α_{i} : i < λ} with α_{i} increasing continuously with i. Notice that for every i, ξ < ζ < μ ⇒ f^{ξ}(α_{i}) ≦ f^{ζ}(α_{i}). Let E^{∗} = {i : α_{i} = i}.

We now define M̄ = ⟨M_{j} : j < λ⟩ by defining M_{j} by induction on j. If j = α_{j} ∈ E^{∗} let M_{j} = ⋃_{ξ<μ} M_{j}^{ξ}. If α < j ≤ α + f^{ξ}(α) for some α ∈ E^{∗} and some ξ < μ, let

ξ′ = sup{ξ : α + f^{ξ}(α) ≤ j} and set M_{j} = ⋃_{ξ>ξ′} M_{j}^{ξ}. If j = sup_{ξ}(α + f^{ξ}(α)) for some

α ∈ E and j > α + f^{ξ}(α) for each ξ < μ, let M_{j} = ⋃_{β<j} M_{β}. Finally, if j < λ does

not fall under any of the previous cases, let M_{j} = ⋃_{ξ<μ} M_{α_{j}}^{ξ}.

We claim that M̄ ∈ Seq_{λ}. One checks that M̄ is continuous and increasing, the main point being that if α ∈ E^{∗} and α < j_{1} < α + f^{ξ_{1}}(α) ≤ j_{2} < α + f^{ξ_{2}}(α) with ξ_{1} < ξ_{2} < μ, then M_{j_{1}}^{ξ_{1}} ≤ M_{j_{1}}^{ξ_{2}} ≤ M_{j_{2}}^{ξ_{2}}. One must also check that ⟨M_{j}, M_{j+1}⟩ ∈ Seq_{2} for all j. This follows from clause (c) of Definition 3.4(7). Let f be defined by f(α_{i}) = sup{f^{ξ}(α_{i}) : ξ < μ} if i ∈ E^{∗} and f(α_{i}) = 0 otherwise. Clearly (M̄^{ξ}, f^{ξ}) ≤ (M̄, f) for ξ < μ.

What about being a ≤-m.u.b.? Assume that (M̄′, f′) ∈ K_{λ}^{qr} and ξ < μ ⇒ (M̄^{ξ}, f^{ξ}) ≤ (M̄′, f′). So for each ξ < μ some club E′_{ξ} of λ exemplifies Definition

3.4(4), and let E′ =: ⋂_{ξ<μ} E′_{ξ} ∩ E^{∗}, a club of λ.

Now for δ ∈ E′ we have (∀ξ < μ)(f^{ξ}(δ) ≤ f′(δ)) hence f(δ) = sup_{ξ<μ}f^{ξ}(δ) ≤ f′(δ),

so δ ∈ E′ ⇒ f(δ) ≤ f′(δ). Similarly δ ∈ E′ & j ≤ f(i) ⇒ M_{δ+j} ≤_{𝔎} M′_{δ+j}. So clearly (M̄, f) ≤ (M̄′, f′) and (M̄, f) is a minimal u.b. (see Definition 3.4(10)).

4) As in the proof of part (3), let ⟨(M̄^{ξ}, f^{ξ}) : ξ < μ⟩ be as therein and let (M̄, f) be constructed as above. For proving it is a lub, let (M^{ξ}, f^{ξ}) ≤ (M̄′, f′) for ξ < μ, and define E′ as there. For δ ∈ E′, j < f(δ) we have ⟨(M_{δ+j}^{ξ}, M_{δ+j+1}^{ξ}) : ξ ∈ (ξ_{δ,j+1}, μ)⟩ is ≤_{{[0,1]}}^{∗}-increasing continuous and ⟨M_{δ+j}^{ξ}, M_{δ+j+1}^{ξ}⟩ ≤_{{[0,1]}}^{∗} (M′_{δ+j}, M′_{δ+j+1}) for ξ ∈ (ξ_{δ,j+1}, μ), so as C is explicitly local by clause (d) in Definition 3.4(11) we have

⟨M_{δ+j}, M_{δ+j+1}⟩ = ⟨ ⋃_{ξ∈(ξ_{δ,j+1},μ)} M_{δ+j}^{ξ}, ⋃_{ξ∈(ξ_{δ,j+1},μ)} M_{δ+j+1}^{ξ}⟩ ≤_{{[0,1]}} (M′_{δ+j}, M′_{δ+j+1})

is as required. The proof for the case μ = λ is similar, using diagonal intersection.

5) Left to the reader. □_{3.5}

∗  ∗  ∗

It may clarify matters if we introduce some natural cases of C. We shall use the forthcoming C^{0} in our construction of many models in 𝔎_{λ^{+2}}.

3.6 Definition. For ℓ ∈ {0, 1, 2} and λ = cf(λ) > LS(𝔎), let C = C_{𝔎,λ}^{ℓ} consist of

(a) τ^{+} = τ, 𝔎^{+} = {M ∈ 𝔎_{<λ} : if λ is a successor cardinal then ∥M∥^{+} = λ} (with =^{M} being equality)

(b) Seq_{α} = {M̄ : M̄ = ⟨M_{i} : i < α⟩ is a ≤_{𝔎}-increasing continuous sequence of members of 𝔎_{<λ} and if α = λ then ⋃_{i<α} M_{i} has cardinality λ} (c) M̄ <_{t}^{∗} N̄ when:

(α) M̄ = ⟨M_{i} : i < α^{∗}⟩, N̄ = ⟨N_{i} : i < β^{∗}⟩ are from Seq (β) if [γ_{1}, γ_{2}] ∈ t then:

(i) γ ∈ [γ_{1}, γ_{2}] ⇒ M_{γ} ≤_{𝔎} N_{γ}

(ii) if ℓ = 1, then in addition γ ∈ [γ_{1}, γ_{2}) ⇒ M_{γ} = M_{γ+1} ∩ N_{γ} (iii) if ℓ = 2, then in addition

γ ∈ [γ_{1}, γ_{2}) ⇒ M_{γ} = M ∩ N_{γ} where M = ⋃_{i<α^{∗}} M_{i}.

Note that C_{𝔎,λ}^{1} is interesting when we have disjoint amalgamation in the appropriate cases.

3.7 Fact.

1) If ℓ = 0 or 1, then C_{𝔎,λ}^{ℓ} is an explicitly local λ-construction framework, (hence λ-nice by 3.5(5)) and 𝔎^{+} satisfies axioms I-VI.

2) If ℓ = 0 or 2, then C_{𝔎,λ}^{ℓ} is an explicitly revised local λ-construction framework (hence λ-nice by 3.5(5)) and 𝔎^{+} satisfies axioms I-VI.

3.8 Definition. If λ = cf(λ) > LS(𝔎) then C_{𝔎,λ}^{3} consists of

(a) τ^{+} = τ ∪ {P, <}, 𝔎^{+} is the set of (M, P^{M}, <^{M}) where M ∈ 𝔎_{<λ}, P^{M} ⊆ M, <^{M} a linear ordering of P^{M} (but =^{M} may be as in 3.1(2)) and M_{1} ≤_{𝔎^{+}} M_{2} iff (M_{1} ↾ τ) ≤_{𝔎} (M_{2} ↾ τ) and M_{1} ⊆ M_{2}

(b) Seq_{α} = {M̄ : M̄ = ⟨M_{i} : i ≤ α⟩ is an increasing continuous sequence of members of 𝔎^{+} and ⟨M_{i} ↾ τ : i ≤ α⟩ is ≤_{𝔎}-increasing, and for i < j < α : P^{M_{i}} is a proper initial segment of (P^{M_{j}}, <^{M_{j}}) and there is a first element in the difference} we denote the <^{M_{i+1}}-first element of P^{M_{i+1}}\P^{M_{i}}, by a_{i}[M̄]

(d) M̄ <_{t}^{∗} N̄ \underline{iff} M̄ = ⟨M_{i} : i < α^{∗}⟩, N̄ = ⟨N_{i} : i < α^{∗∗}⟩ are from Seq, t is a set of pairwise disjoint closed intervals of α^{∗} and for any [α, β] ∈ t we have (β < α^{∗} and): γ ∈ [α, β] ⇒ M_{γ} ≤_{𝔎} N_{γ} & a_{γ}[M̄] ∉ N_{γ}, moreover a_{γ}[M̄] = a_{γ}[N̄].

3.9 Fact. C_{𝔎,λ}^{3} is an explicitly local and revised local λ-construction framework (hence λ-nice by 3.5(5)) and 𝔎^{+} satisfies axioms I-VI.

We now introduce amalgamation choice functions. The use of “F a λ-amalgamation choice function” is to help use the weak diamond, by taking out most of the freedom in choosing amalgams. This gives possibilities for coding (3.14, 3.17).

3.10 Definition. 1) We say that F is a λ-amalgamation choice function for the construction framework C \underline{if} F is a five place function satisfying:

(a) if M_{ℓ} ∈ K_{<λ}^{+} for ℓ < 3, M_{0} ≤_{𝔎^{+}} M_{1}, M_{0} ≤_{𝔎^{+}} M_{2}, M_{1} ∩ M_{2} = M_{0} (before dividing by =^{M_{ℓ}}); a ∈ M_{2} and (∀b ∈ M_{0})[¬a =^{M_{1}} b]; and A is a set such that A ∪ |M_{1}| ∪ |M_{2}| is a set of ordinals \underline{then} F (M_{0}, M_{1}, M_{2}, A, a), if defined, is a member N of 𝔎^{+} with universe A ∪ |M_{1}| ∪ |M_{2}|, which ≤_{𝔎^{+}}-extends M_{1} and M_{2} and (a/ =^{N}) ∉ (M_{1}/ =^{N});

(b) [uniqueness] if (M_{0}, M_{1}, M_{2}, A, a) and (M′_{0}, M′_{1}, M′_{2}, A′, a′) are as above and in the domain of F, f is an order preserving mapping from A ∪ |M_{1}| ∪ |M_{2}| onto A′ ∪ |M′_{1}| ∪ |M′_{2}| such that f ↾ M_{ℓ} is an isomorphism from M_{ℓ} onto M′_{ℓ} for ℓ = 0, 1, 2 (so preserving =^{M_{ℓ}}, and its negation) and f(a) = a′ \underline{then} f is an isomorphism from F (M_{0}, M_{1}, M_{2}, A, a) onto F (M′_{0}, M′_{1}, M′_{2}, A′, a′);

(c) if F (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) is well defined then x_{0}, x_{1}, x_{2}, x_{3}, x_{4} are as in part (a).

Observe that as M_{1} ∩ M_{2} = M_{0} in (a), if we do not have disjoint amalgamation then we are forced to allow =_{N} to be a nontrivial congruence.

2) If F is defined whenever the conditions in part (a) hold and A\M_{1}\M_{2} has large enough cardinality \underline{then} we say F is full (if λ = μ^{+}, it suffices to demand A\M_{1}\M_{2} has cardinality μ).

3) We say F has strong uniqueness \underline{if}

(d) if (M_{0}, M_{1}, M_{2}, A, a) and (M′_{0}, M′_{1}, M′_{2}, A′, a′) are as above and in the domain of F and for ℓ = 0, 1, 2 we have R_{ℓ} is a isomorphism relation from M_{ℓ} onto M′_{ℓ} such that R_{0} = R_{1} ∩ (M_{0} × M′_{0}) = R_{2} ∩ (M_{0} × M′_{0}) and |A\M_{1}\M_{2}| = |A′\M′_{1}\M′_{2}|, \underline{then} there is an isomorphism relation R from M = F (M_{0}, M_{1}, M_{2}, A, a) onto M′ = F (M′_{0}, M′_{1}, M′_{2}, A′, a) such that R_{ℓ} = R ∩ (M_{ℓ} × M′_{ℓ}).

3.11 Definition. Assume C is a λ-construction framework and F is a λ-amalgamation choice function for C. Let (M̄^{ℓ}, f^{ℓ}) ∈ K_{λ}^{qr} for ℓ = 1, 2.

1) (M̄^{1}, f^{1}) <_{F,a}^{at} (M̄^{2}, f^{2}) (if we omit a, this means for some a; “at” stands for atomic extension; we may write ≤_{F,a}^{at} instead of <_{F,a}^{at}) means that:

(a) (M̄^{1}, f^{1}) ≤ (M̄^{2}, f^{2})

(b) for some club E of λ, for every δ ∈ E taking e_{δ} =: [δ, δ + f^{1}(δ)] we have

(∗) if β < γ are successive members of e_{δ} then: M_{γ}^{2} = F (M_{β}^{1}, M_{γ}^{1}, M_{β}^{2}, |M_{γ}^{2}|, a)

(∗∗) |M_{γ}^{2}| = |M_{γ}^{1}| ∪ |M_{β}^{2}| ∪ {i ∈ M_{γ}^{2} : i an ordinal not in |M_{γ}^{1}| ∪ |M_{β}^{2}| and otp(|M_{γ}^{2}| ∩ i\|M_{γ}^{1}|\|M_{β}^{2}|) < ∥M_{γ}^{2}∥

(∗ ∗ ∗) ∥M_{γ}^{2}∥ = Min{∥N∥ : N = F (M_{β}^{1}, M_{δ}^{1}, M_{β}^{2}, |N|, a)}.

A suitable club E may be called a witness for the relation. Implicit in clause (b) in a ∈ M^{2} and ¬(∃b)(b ∈ M^{1} & a =^{M^{2}} b).

2) (M̄′, f′) ≤_{F} (M̄′′, f′′) means that: there is a sequence ⟨(M̄^{ζ}, f^{ζ}) : ζ ≤ ξ⟩ such that:

(a) ξ < λ

(b) (M̄^{ζ}, f^{ζ}) ∈ K_{λ}^{qr} is ≤-increasing continuous in ζ (remember Definition 3.4(10))

(c) for each ζ < ξ we have (M̄^{ζ}, f^{ζ}) ≤_{F}^{at} (M̄^{ζ+1}, f^{ζ+1})

(d) (M̄′, f′) = (M̄, f^{0}) and (M̄′′, f′′) = (M̄^{ξ}, f^{ξ}).

A club E which witnesses all the relations in (c) or at least each of them is witnessed by some end segment of E, is called a witness for the relation ≤_{F}.

3) <_{F,a}^{at,∗} is defined similarly to part (1) but we demand in clause (b) only that e_{δ} ⊆ [δ, δ + f^{2}(δ)] is closed and {δ, δ + f^{2}(δ)} ⊆ e_{δ}; the requirement from clause (a) is unchanged and we require also:

if β ∈ [δ, δ + f^{2}(δ)] then M_{β}^{2} = M_{max(e_{δ}∩(β+1))}^{2}.

Then define ≤_{F}^{∗} by iterating ≤_{F}^{at,∗}.

4) We may replace F by F, a family of such functions. Then in each case in 3.11(2)(c) we use one such that F ’s. F^{∗} is the family of all such F ’s.

5) (M̄^{1}, f^{1}) <_{F,a} (M̄^{2}, f^{2}) means that for some (M̄, f) we have (M̄^{1}, f^{1}) ≤_{F,a}^{at} (M̄, f̄) ≤_{F} (M̄^{2}, f^{2}).

6) We define mub as in 3.4(10).

3.12 Remark. 1) What we prove below on <_{F,a}^{at}, ≤_{F} also holds for <_{F,a}^{at,∗}, ≤_{F}^{∗}.

2) Note: using F rather than F may help in proving cases of Definition 3.20, but we can use one F which codes all members of F by asking on A\M_{1}\M_{2}, though artificially.

3) We can replace F by ⟨F_{η} : η a sequence of ordinals of length < λ, η(1 + i) < 2, η(0) < 2^{<λ}⟩, each F_{η} with uniqueness 3.10(3) and (∗∗) of 3.11(1)(b) is replaced by M_{γ}^{2} = f_{η↾δ}(M_{β}^{1}, M_{γ}^{1}, M_{β}^{2}, |M_{γ}^{2}|, a), and omit (∗ ∗ ∗) there.

3.13 Claim. If C is nice, \underline{then} on K_{λ}^{qr}, ≤_{F} is a quasi-order, and every increasing continuous sequence of length less than λ^{+} has a mub.

Proof. Check. □_{3.13}

3.14 Definition. 1) We say a λ-amalgamation choice function F for 𝔎^{+} has the λcoding property for C if: Seq_{λ} ≠ ∅, and for every M̄^{1} ∈ Seq_{λ}, function f^{1} : λ → λ, and S ⊆ λ we can find M̄^{2,η} ∈ Seq_{λ} for η ∈ ^{λ}2 with η extending 0_{λ\S} that is η ↾ (λ\S) being constantly zero, a function f^{2} : λ → λ such that f^{1} ≤_{𝒟_{λ}} f^{2}, f^{2} ↾ (λ\S) = f^{1} ↾ (λ\S) and an element a_{η} of M^{2,η} (usually a_{η} = a) such that:

(∗)_{1} (M̄^{1}, f^{1}) ≤_{F,a}^{at} (M̄^{2,η}, f^{1}) for all η extending 0_{λ\S}, and

η ↾ α = ν ↾ α ⇒ M^{2,η} ↾ α = M^{2,ν} ↾ α; (∗)_{2} for some club E of λ the following is impossible: for some η_{3}, η_{4} ∈ ^{λ}2

extending 0_{λ\S}, for ℓ = 3, 4 we have (M̄^{1}, f^{2}) ≤_{F,a^{ℓ}} (M̄^{ℓ}, f^{ℓ}) witnessed by a club E_{ℓ} we have (abusing our notation we are dividing by the equality congrugence) f_{ℓ} a ≤_{𝔎}-embedding of M^{2,η_{ℓ}} ↾ τ into M^{ℓ} ↾ τ over M_{λ}^{1}, and for some δ ∈ E ∩ E_{3} ∩ E_{4} ∩ S we have M̄^{3} ↾ δ = M̄^{4} ↾ δ, a^{3} ∈ M_{δ}^{2,η_{3}}, a^{4} ∈ M_{δ}^{2,η_{4}}, f_{3}(a^{3}) = f_{4}(a^{4}), f^{3} ↾ δ = f^{4} ↾ δ, f_{3} ↾ M_{δ}^{2,η_{3}} = f_{4} ↾ M_{δ}^{2,η_{4}}, and ℓ ∈ {3, 4} ⇒ Rang(f_{ℓ} ↾ M^{2,η_{ℓ}}) ⊆ M_{δ}^{ℓ}, η_{3} ↾ δ = η_{4} ↾ δ, η_{3}(δ) ≠ η_{4}(δ).

2) We say that F has the weak λ-coding property if above we restrict ourselves to the cases f^{1} ↾ S = 0_{S}. We can even restrict ourselves to the cases f^{1} ∈ ℱ ⊆ ^{λ}λ provided that 0_{λ} ∈ ℱ and we demand f^{2} ∈ ℱ.

3) If we replace F by F, a family of such functions it means we use Definition 3.11(4).

4) We say 𝔎 has a coding property if some λ-amalgamation choice function F has this property. Typically the actual choice of F is irrelevant as long as its domain is sufficiently rich.

3.15 Observation. The restriction above to η such that η extends 0_{λ\S}, S is natural but inessential, as we can extend the definition of M^{2,η} to all η in ^{λ}2 by defining M^{2,η} = M^{2,η′} where η′ ↾ S = η ↾ S and η′ ↾ (λ\S) is constantly zero. Then the same properties will hold.

3.16 Remark.

1) For a local construction framework C in 3.14(1) the conditions (∗)_{1} and (∗)_{2} can be replaced by local requirements. For example, in condition (∗)_{1}, we may take 3.4(7b) into account.

2) In 3.14(∗)_{2} a sufficient condition for the impossibility of the stated conditions on E, η^{3}, η^{4} and δ, where η^{3} ↾ δ = η^{4} ↾ δ = η, say, is that there is ā ∈ M_{δ}^{2,η} so that:

(∗)_{3} tp(ā, M_{δ+f^{2}(δ)}^{1}, M^{2,η^{3}}) ≠ tp(ā, M_{δ+f^{2}(δ)}^{1}, M^{2,η^{4}}).

We can even allow ā to be infinite here, say a full listing of M_{δ}^{2,η}.

To see that this suffices, suppose that we also have the conditions of 3.14(∗)_{2}. Then for ℓ = 3, 4 as M_{δ+j+1}^{2,η_{ℓ}} is given by

F (M_{δ+j}^{1}, M_{δ+j+1}^{1}, M_{δ+j}^{2,η_{ℓ}}, |M_{δ+j+1}^{2,η_{ℓ}}|, a^{ℓ})

where j < f^{2}(δ), we find M_{δ+f^{2}(δ)}^{2,η_{3}} = M_{δ+f^{2}(δ)}^{2,η_{4}} = M^{∗}, say. Since M^{2,η_{3}} and M^{2,η_{4}} can be amalgamated over M^{∗}, we have

tp(f_{3}(ā), M_{δ+f^{2}(δ)}^{1}, M^{3,η_{3}}) = tp(f_{4}(ā), M_{δ+f^{2}(δ)}^{1}, M^{4,η_{4}}).

On the other hand by (∗)_{2} we have

tp(ā, M_{δ+f^{2}(δ)}^{1}, M^{2,η_{ℓ}}) = tp(f_{ℓ}(ā), M_{δ+f^{2}(δ)}^{1}, M^{ℓ,η_{ℓ}})

and this gives a contradiction.

3) To understand Definitions 3.14(1,2) you may look at the places they are verified, such as 3.25 and 3.27. Also see 3.21(3,4).

4) The next lemma deduces from the criterion in 3.14(1) another one which is natural for use in a non-structure theorem.

5) Note that 3.14(1) implies: for every (M̄^{1}, f^{1}) there is (M̄^{2}, f^{2}) such that (M̄^{1}, f^{1}) ≤_{F,a}^{at} (M̄^{2}, f^{2}) and M_{λ}^{1} ≠ M_{λ}^{2}.

6) In Definition 3.14(1) we can replace <_{F,a}^{at} by <_{F} or <_{F,a} with no harm as <_{F,a} satisfies the requirement on <_{F,a}^{at} and starting from it we again get <_{F}.

7) In (∗)_{2} of Definition 3.14(1) for some function H depending on (M̄^{1}, f^{1}) we may add the further restriction ℓ ∈ {3, 4} and α < δ implies δ ∈ H(M̄^{ℓ} ↾ α, f_{ℓ} ↾ M_{α}^{ℓ}). I.e. this weakening of the demand does not change the desired conclusions.

8) We can weaken the demand in (∗)_{2} of 3.14(1) to extensions which actually arise but this seems more cumbersome. While the adaptation is straightforward, we have no application in mind.

9) In 3.14(1), (∗)_{2} we may strengthen the requirement by excluding the case where the club E is allowed to depend on η. That is, we consider quadruples (E^{η}, M̄^{η}, f^{η}, f^{η}) for η ∈ ^{λ}2 such that (M̄^{1}, f^{2}) ≤_{F} (M̄^{η}, f^{η}) is witnessed by a club E^{η} in λ and f^{η} is a ≤_{𝔎}-embedding of M^{2,η} ↾ τ into M^{η} ↾ τ over M^{1}. We require:

(∗)′_{2} for no η^{3}, η^{4} ∈ ^{λ}2 and δ ∈ E^{η^{3}} ∩ E^{η^{4}} ∩ E ∩ S do we have:

η_{3} ↾ δ = η_{4} ↾ δ, η_{3}(δ) ≠ η_{4}(δ), f^{3} ↾ δ = f^{4} ↾ δ;

M̄^{2,η_{3}} ↾ [δ, δ + f^{2}(δ)] = M̄^{2,η_{4}} ↾ [δ, δ + f^{3}(δ)];

f^{η_{3}} ↾ M_{δ}^{2,η_{3}} = f^{η_{4}} ↾ M_{δ}^{2,η_{4}};

f^{η_{ℓ}}[M_{δ}^{2,η_{ℓ}}] ⊆ M_{δ}^{η_{ℓ}} for ℓ = 3, 4.

10) In 3.14 we can also require the models M^{1}, M^{2,η} to have universes λ(1+α) and λ(1 + α + 1) respectively for some α, with λ(1 + α) ∈ M_{0}^{2,η}. This will not change much.

3.17 Lemma. Assume (∃μ)(λ = μ^{+} & 2^{μ} < 2^{μ^{+}}) or at least the definitional weak diamond on λ holds. Assume C is λ-nice, J = WDmId^{Def}(λ). If the λ-amalgamation choice function F has the λ-coding property, \underline{then} it has the explicit (λ, J)-coding property, which means: if (M̄^{1}, f̄^{1}) ∈ K_{λ}^{qs} and S ⊆ λ satisfies S ∉ J \underline{then} we can find (M̄^{2}, f^{2}) ∈ K_{λ}^{qs} such that:

(a) (M̄^{1}, f^{1}) ≤_{F}^{at} (M̄^{2}, f^{1}) and f^{1} ↾ (λ\S) = f^{2} ↾ (λ\S)

(b) if (M̄^{1}, f^{2}) ≤_{F} (M̄^{3}, f^{3}) ∈ K_{λ}^{qs} \underline{then} M^{2} ↾ τ cannot be ≤_{𝔎}-embedded into M^{3} ↾ τ over M^{1}.

Proof. The proof is straight forward once you digest the meaning of weak diamond.

Let S = λ. Suppose ⟨M̄^{2,η} : η ∈ ^{λ}2⟩ and f^{2} are as in 3.14(1), taking note of 3.15. Then we claim there is ν ∈ ^{λ}2 for which 3.17 holds on taking M̄^{2} to be M̄^{2,ν}. Assume toward a contradiction that this fails for each ν. Then clause (b) fails, and for each ν ∈ ^{λ}2 we have some M̄^{3,ν} and f^{3,ν} with:

(M̄^{1}, f^{2}) ≤_{F} (M̄^{3,ν}, f^{3,ν}) witnessed by a club E^{ν};

f_{ν} : M^{2,ν} ↾ τ → M^{3,ν} ↾ τ over M^{1} ↾ τ a ≤_{𝔎} -embedding.

Now by the definition of WDmId^{Def}(λ) we can find η_{3} ≠ η_{4} in ^{λ}2 and δ ∈ E^{η_{3}} ∩ E^{η_{4}} ∩ E^{∗} ∩ S with E^{∗} = {α < λ limit: β < α implies β + f^{2}(δ), β + f^{1}(δ) < α}, as forbidden in (∗)_{2} of 3.14(1). □_{3.17}

Now we can give a reasonable non-structure theorem.

3.18 Theorem. Assume C is λ-nice, (∃μ)(λ = μ^{+} & 2^{μ} < 2^{μ^{+}}) & 2^{λ} < 2^{λ^{+}}, or at least DWD(λ), and DWD^{+}(λ^{+}). Let J = WDmId^{Def}(λ).

If F has the (λ, J)-coding property, \underline{then} I(λ^{+}, 𝔎) ≥ 2^{λ^{+}}; also if 2^{λ^{+}} > (2^{λ})^{+} then IE(λ^{+}, 𝔎) ≥ 2^{λ^{+}}.

Proof. We choose by induction on α < λ^{+}, for every η ∈ ^{α}2, a pair (M̄^{η}, f^{η}) such that:

(a) (M̄^{η}, f^{η}) ∈ K_{λ}^{qs}

(b) if ν ◁ η then (M̄^{ν}, f^{ν}) ≤_{F} (M̄^{η}, f̄^{η})

(c) (M̄^{η}, f^{η}) ≤_{F,a_{ηˆ ⟨0⟩}}^{at} (M̄^{ηˆ ⟨0⟩}, f^{ηˆ ⟨0⟩})

(d) (M̄^{η}, f̄^{η}) ≤_{F,a_{ηˆ ⟨1⟩}}^{at} (M̄^{ηˆ ⟨1⟩}, f^{ηˆ ⟨1⟩})

(e) if ℓg(η) is a limit ordinal and (M̄^{η}, f̄^{ηˆ ⟨0⟩}) ≤_{F} (M̄′, f′) \underline{then} M^{ηˆ ⟨1⟩} ↾ τ cannot be ≤_{𝔎}-embedded into M′ ↾ τ over M^{η}

(f) if α is limit ordinal, then (M̄^{η}, f^{η}) is a ≤_{F} -mub of ⟨(M̄^{η↾i}, f^{η↾i}) : i < α⟩.

For α = 0 note that as Seq_{λ} ≠ ∅ also Seq_{λ}^{s} ≠ ∅ hence K_{λ}^{qs} ≠ ∅. For α limit use 3.6, 3.10. For α = β + 1, β a limit ordinal and ν ∈ ^{β}2, define (M̄^{νˆ ⟨ℓ⟩}, f^{νˆ ⟨ℓ⟩}) for ℓ = 0, 1 by 3.17. If α = β + 1, β non-limit use 3.16(3). Let M^{η} = ⋃_{α<λ^{+}} M^{η↾α} ↾ τ for

η ∈ ^{(λ^{+})}2. Now note {a_{η↾(i+1)}/ =^{M^{η}}: i < λ^{+}} ⊆ M^{η}/ =^{M^{η}} are pairwise distinct so M^{η} ∈ K_{λ^{+}}. Now we can apply 1.6 (with λ^{+} here standing for λ there). □_{3.18}

∗  ∗  ∗

Unfortunately, in some interesting cases we get only weak coding.

3.19 Theorem. Assume C is λ-nice, (∃μ)(λ = μ^{+} and 2^{μ} < 2^{λ} < 2^{λ^{+}} and WDmId(λ^{+}) is not λ^{+}-saturated (or at least DfWD(λ), DfWD^{+}(λ^{+}) and WDmId^{Def}(λ) is not λ^{+}-saturated)). If F has the weak λ-coding property (see Definition 3.14(2)), or at least the parallel of the conclusion of 3.17, \underline{then} I(λ^{+}, K) ≥ 2^{λ^{+}}.

Proof. We can find ⟨S_{α}^{∗} : α < λ^{+}⟩ such that:

S_{α}^{∗} ⊆ λ

[α < β ⇒ |S_{α}^{∗}\S_{β}^{∗}| < λ]

and

S′_{α} =: S_{α+1}^{∗}\S_{α}^{∗} ∉ WDmId^{def}(λ).

We again choose by induction on α < λ^{+} for every η ∈ ^{α}2 a pair (M̄^{η}, f^{η}) such that:

(a) (M̄^{η}, f^{η}) ∈ K_{λ}^{qs}

(b) if ν ◁ η, \underline{then} (M̄^{ν}, f^{ν}) ≤_{F} (M̄^{η}, f^{η})

(c) f^{η} ↾ (λ\S_{ℓg(η)}) = 0_{λ\S_{ℓg(η)}}

(d) (M̄^{η}, f^{η}) <_{F,a_{ηˆ ⟨0⟩}}^{at} (M̄^{ηˆ ⟨0⟩}, f^{ηˆ ⟨0⟩})

(e) (M̄^{η}, f^{η}) <_{F,a_{ηˆ ⟨1⟩}}^{at} (M̄^{ηˆ ⟨1⟩}, f^{η})

(f) if (M̄^{η}, f^{ηˆ ⟨0⟩}) ≤_{F} (M̄′, f′) \underline{then} M^{ηˆ ⟨1⟩} ↾ τ cannot be ≤_{𝔎}-embedded into M′ ↾ τ over M^{η}

(g) if α is a limit ordinal, \underline{then} (M̄^{η}, f^{η}) is a ≤_{F} -mub of ⟨(M̄^{η↾i}, f^{η↾i}) : i < α⟩.

Again there are no problems (the difference is in clause (c)). Then we apply 1.6(1) (or 1.7(1)). □_{3.19}

3.20 Definition. 1) We say a λ-amalgamation choice function F for C has the weak^{3} (or weakly) local λ-coding property for C if:

(∗)_{1} Assume ⟨M_{0}, M_{1}⟩ ∈ Seq_{2}, M_{0} ≤_{𝔎^{+}} N_{0} ∈ 𝔎_{<λ}^{+}, M_{1} ∪ N_{0} ⊆ λ^{+}, and |M_{1}| ∩ |N_{0}| = |M_{0}| and a ∈ N_{0}, a/ =^{N_{0}} ∉ (M_{0}/ =^{N_{0}}) (i.e. (∀b ∈ M_{0})(¬a =^{N_{0}} b)).

\underline{Then} we can find N^{1}, N^{2} ∈ 𝔎_{<λ}^{+} such that:
(a) N^{1} = F (M_{0}, M_{1}, N_{0}, |N^{1}|, a)
(b) ⟨N_{0}, N^{1}⟩ ∈ Seq_{2} and ⟨N_{0}, N^{2}⟩ ∈ Seq_{2}
(c) M_{1} ≤_{𝔎^{+}} N^{1} and M_{1} ≤_{𝔎^{+}} N^{2}

(d) N^{1} ↾ τ, N^{2} ↾ τ are contradicting^{4} amalgamations of M_{1} ↾ τ, N_{0} ↾ τ over M_{0} ↾ τ; i.e. for no N′, h do we have: (N^{1} ↾ τ) ≤_{𝔎} N′ ∈ 𝔎_{<λ} and

h is a ≤_{𝔎}-embedding of N^{2} into N′ over M_{1} ∪ N_{0}; or at least

(d)^{−} (N^{1}, N^{2}) is a τ-contradicting pair of amalgamations of M_{1}, N_{0} over M_{0} which just says: if N^{1} ≤_{𝔎^{+}} N ∈ 𝔎_{<λ}^{1} then there is no ≤_{𝔎}-embedding h of N^{2} ↾ τ into N ↾ τ over M_{1} ∪ N_{0} (i.e. is the identity on M_{1} and

on N_{0})

(note: this is not necessarily symmetric; and we use just the τ-reducts of N^{2}, M_{0}, M_{1}, N_{0} so we can replace them by N^{2} ↾ τ, M_{0} ↾ τ, M_{1} ↾ τ, N_{0} ↾ τ respectively).

2) We say that a λ-amalgamation choice function F for C has the local λ-coding property if:

3there is no clear relation between “weak local” and “local” λ-coding inspite of the name

4Of course, we may consider only ones in “legal” extensions. We can also note that for the intended use, the disjointness is automatic (so 3.1(2) not needed)

(∗)_{2} if M̄ = ⟨M_{j} : j < λ⟩ ∈ Seq_{λ}, N̄ = ⟨N_{j} : j ≤ δ + i⟩ ∈ Seq_{δ+i}, a ∈ N_{0}, (a/ =^{N_{δ+i}}) ∉ M_{δ+i}/ =^{M_{δ+i}} and M̄ ↾ (δ + i + 1) ≤_{{[δ,δ+i]}}^{∗} N̄, and N_{δ+j+1} = F (M_{δ+j}, M_{δ+j+1}, N_{δ+j}, |N_{δ+j+1}|, a) for j < i \underline{then} for some i_{1}, i_{2} ∈ (i, λ) and N̄^{ℓ} = ⟨N_{α}^{ℓ} : α < δ + i_{ℓ}⟩ ∈ Seq_{δ+i_{ℓ}} for ℓ = 1, 2 we have:

(a) N̄^{ℓ} ↾ (δ + i + 1) = N̄ for ℓ = 1, 2

(b) for j ∈ [δ + i, δ + i_{1}) we have N_{j+1}^{1} = F (M_{j}, M_{j+1}, N_{j}^{1}, |N_{j+1}^{1}|, a)

(c) M_{δ+i_{2}} ≤_{𝔎^{+}} N_{δ+i_{2}}^{2}

(d) N_{δ+i_{1}}^{1} ↾ τ, N_{δ+i_{2}}^{2} ↾ τ are contradictory amalgamations of M_{δ+i_{1}} ↾ τ and N_{δ} ↾ τ over M_{δ} ↾ τ or at least

(d)^{−} (N_{δ+i_{1}}^{1}, N_{δ+i_{2}}^{2}) are τ-contradictory amalgamations of M_{δ+i(∗)} and N_{δ} over M_{δ} where i(∗) = Min{i_{1}, i_{2}}.

(So if i = 0, this gives us a possibility to amalgamate, helpful for

i ∈ λ\ ⋃_{δ∈E}[δ, δ + f(δ))).

3) We say that a λ-amalgamation choice function F for C has the weaker local λ-coding property for C if:

(∗)_{3} as in part (2) but i = 0.

4) In 1), 2), 3) above we say C has weak local or the local or the weaker local coding property respectively, if we omit the mention of F meaning for some F (clause (a) in (∗)_{1}, clause (b) in (∗)_{2}.)

3.21 Claim. 1) If F has the weak local λ-coding property for C or F has the local λ-coding property for C \underline{then} F has the weaker local λ-coding property for C.

2) Assume

(a) C has a local λ-construction framework

(b) F has the λ-coding (or weaker λ-coding) (or the weak λ-coding) property for C.

\underline{Then} for some F′ we have:

(α) F′, too, is a λ-amalgamation choice function for C

(β) if F (N_{0}, N_{1}, N_{2}, A, a) is well defined and its τ-reduct is <_{𝔎} M ∈ K_{<λ} and A ⊆ A′ ⊆ λ^{+}, |A′| < λ, \underline{then} for some A′′, A′ ⊆ A′′ ⊆ λ^{+}, |A′′| < λ, and F′(N_{0}, N_{1}, N_{2}, A′, a) is well defined and F (N_{0}, N_{1}, N_{2}, A, a) ≤_{𝔎^{+}} F′(N_{0}, N_{1}, N_{2}, A′′, a) and M ≤_{𝔎} F′(N_{0}, N_{1}, N_{2}, A′′, a) ↾ τ

(γ) F′ has the local or weak local or weaker local (respectively as in (b)) λ-coding property for C.

3) If C is local (λ-construction framework), F a λ-amalgamation choice function, with the weak (or just weaker) local λ-coding property, \underline{then} F has the weak λ-coding property (hence under the set theoretic assumptions of 3.19, I(λ^{+}, 𝔎) ≥ 2^{λ^{+}}).

4) If C is local (λ-construction framework), F a λ-amalgamation choice function, with the local λ-coding property, \underline{then} F has the λ-coding property (hence under the set theoretic assumptions of 3.18, we have I(λ^{+}, 𝔎) ≥ 2^{λ^{+}} and (2^{λ})^{+} < 2^{λ^{+}} ⇒ IE(λ^{+}, 𝔎) ≥ 2^{λ^{+}}).

Remark. The parallel of part (2) holds for local and weaker local property \underline{if} F acts on sequences. See 3.22 below.

Proof. 1) Check.

2) Here we use clause (d)^{−} rather than (d).

3) Similar (or read the proof of ?).

4) Check. □_{3.21}

In this context we may consider

3.22 Definition. We say that F is a λ-amalgamation choice function for sequences, for C if:

(a) if x = F (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) is defined then for some α_{1}, α_{2}, α_{3}, α < λ we have x_{ℓ} = M̄^{ℓ} ∈ Seq_{α_{ℓ}}^{s} for ℓ < 3, M̄^{1} ◁M̄^{2}, t = x_{4} is a set of pairwise disjoint intervals ⊆ α_{i}, M̄^{1} <_{t}^{∗} M̄^{3}, A = x_{5} a set of < λ ordinals < λ^{+}, x = M̄ ∈ Seq_{α}^{s}, M_{α} has universe A, M̄^{1} ≤_{t∪{[α_{1},α_{2}]}}^{∗} M̄

(b) [uniqueness] as in Definition 3.10.

3.23 Claim. Assume:

(a) (∃μ < λ)(2^{μ} = 2^{<μ} < 2^{λ}), 2^{λ} < 2^{λ^{+}} or at least we have the definable weak diamond for λ and λ^{+}; and 2^{<λ} ≥ λ^{+}.

(b) C is a nice (λ-construction framework).

(c) F is a λ-amalgamation choice function for C.

(d) F has the weaker λ-coding property for λ.

\underline{Then} I(λ^{+}, 𝔎) ≥ μ_{wd}(λ^{+}).

Proof. Straight forward using 1.4.

Let ⟨M_{i} : i < i^{∗}⟩ list the models in 𝔎_{λ^{+}} up to isomorphisms and assume toward contradiction that i^{∗} < μ_{wd}(λ^{+}). It is enough to choose by induction on α < λ^{+} a sequence M̄_{0}^{η} for η ∈ ^{α}2 such that:

(a) M̄^{η} ∈ Seq_{λ}

(b) M_{λ}^{η} has universe γ_{η} < λ^{+}

(c) ν ◁ η ⇒ M̄^{ν} ≤_{F} M̄^{η} (d) if δ = ℓg(η) is a limit ordinal then M̄^{η} is a mub of ⟨M̄^{η↾α} : α < δ⟩

(e) if ηˆ ⟨0⟩ ⊴ ν ∈ ^{λ^{+}}2 then M_{λ}^{ηˆ ⟨1⟩} ↾ τ cannot be ≤_{𝔎}-embedded into M_{λ}^{ν} ↾ τ over M_{λ}^{η} ↾ τ.

This is possible by 3.21(3). Having the M_{η}, η ∈ ^{λ>}2 we get the conclusion by 1.5. □_{3.23}

3.24 Remark. 1) If we are just interested in I(λ^{+}, K) rather than also in IE(λ^{+}, K), then we can change the definition of τ-contradictory to:

N^{1}, N^{2} are τ-contradictory amalgamations of M_{1}, N over M_{0} \underline{if} M_{0} ≤_{𝔎^{+}} M_{1} ≤_{𝔎^{+}} N^{ℓ}, M_{0} ≤_{𝔎^{+}} N_{0} ≤_{𝔎^{+}} N^{ℓ}, M_{0} = M_{1}∩N_{0} and there are no N_{∗}^{1}, N_{∗}^{2} ∈ 𝔎_{<λ}^{+} such that: N^{ℓ} ≤_{𝔎^{+}} N_{∗}^{ℓ} and N_{∗}^{1} ↾ τ, N_{∗}^{2} ↾ τ are isomorphic over M_{1}∪N_{0}.

2) Note that it is unreasonable to assume that we will always use the local versions: e.g. if we have a superlimit model in 𝔎_{λ^{+}} and we want to have M̄ ∈ Seq_{λ} ⇒ M_{λ} superlimit, we have to add some global condition (see [Sh 600]). Also possibly we will have in the construction (i.e. in 3.18 or 3.19) that M̄_{η} has two “contradictory” extensions M̄_{ηˆ ⟨0⟩}, M̄_{ηˆ ⟨1⟩}, (see clauses (e) and (f) in their proof) only when cf(ℓg(η)) = θ, where S = {δ : δ < λ^{+} and cf(δ) = θ} ∉ WDmId(λ); or even ℓg(η) ∈ S, for a given S ∈ WDmId(λ)^{+}. We shall deal with such cases when needed.

Remark. We intend to continue this elsewhere.

3.25 Lemma. Let 𝔎 be an abstract elementary class with LS(𝔎) ≤ λ which is categorical in λ and in λ^{+}, with 1 ≤ I(λ^{++}, K) < 2^{λ^{++}}. Assume that 2^{λ} < 2^{λ^{+}} < 2^{λ^{++}}, or at least that the definitional weak diamond holds for both λ^{+} and λ^{++}.

If there is a model in K_{λ^{+}} which is saturated over λ, \underline{then} the minimal triples are dense in K_{λ}^{3}.

Proof. Let C be C_{𝔎,λ^{+}}^{0} (see Definition 3.6) hence C is explicitly local λ-construction framework (by 3.7(1)). Suppose toward contradiction that above (M^{∗}, N^{∗}, a^{∗}) ∈ K_{λ}^{3}, there is no minimal triple. We claim in this case that there is a λ^{+}-amalgamation choice function F for C with the λ^{+}-coding property, with domain the quadruples (M_{0}, M_{1}, M_{2}, A, b) such that: (M^{∗}, N^{∗}, a) embeds in (M_{0}, M_{2}, b); A and the universes of M_{1}, M_{2} are contained in λ^{++}; |A\(|M_{1}| ∪ |M_{2}|)| = λ; and M_{0} ≤_{𝔎} M_{1}, M_{2}. Then applying 3.17 and 3.18 we get a contradiction.

We first make two observations concerning triples (M, N, b) lying above (M^{∗}, N^{∗}, b). Any such triple has the extension property by 2.11 (or just 2.9(1)) and hence we can manufacture a λ^{+}-amalgamation choice function with the specified domain. Furthermore, there is M′ ∈ K_{λ^{+}} with M ≤_{𝔎} M′ such that tp(a, M, N) has more than one extension to M′, by the failure of minimality.

Let us show that any λ^{+}-amalgamation choice function F with the specified domain has the λ^{+}-coding property on C = C_{𝔎,λ^{+}}^{0}.

Let M̄^{1} ∈ Seq_{λ^{+}} and let f^{1} : λ → λ. For any set S we must find sequences M̄^{2,η} (depending on S) as in 3.21(1). The approach will be to first build suitable M^{2,η} for all η ∈ ^{λ}2, independent of S, then restrict appropriately given S.

Let M^{1} = ⋃_{i} M_{i}^{1}. Then M^{1} ∈ 𝔎_{λ^{+}} and by our assumptions M^{1} is therefore

saturated over λ. Hence we may suppose M^{∗} ≤_{𝔎} M^{1}. We may also suppose that the universe of M^{1} is an ordinal, and we may choose a subset A^{2} of λ^{++} which is the union of an increasing continuous sequence A_{α}^{2} (for α < λ^{+}) so that: A_{α}^{2} ∩ |M^{1}| = |M_{α}^{1}| and A_{0}^{2}\|M_{0}^{1}| and A_{α+1}^{2}\(A_{α}^{2} ∪ |M_{α+1}^{1})) have cardinality λ. Let E be the club:

{δ < λ : for α < δ, we have α + f^{1}(α) + 1 < δ}.

We now define triples (M_{η}^{∗}, N_{η}^{∗}, a^{∗}) for all η ∈ ^{i}2, by induction on i, together with ordinals α_{η} satisfying the following conditions:

(a) for any η ∈ ^{λ}2, the sequence (M_{η↾j}^{∗}, N_{η↾j}^{∗}, a^{∗})(j ≤ i) is increasing and continuous; and similarly the α_{η} are increasing and continuous.

(b) (M_{<>}^{∗}, N_{<>}^{∗}) = (M^{∗}, N^{∗}).

(c) M_{η}^{∗} = M_{α_{η}}^{1} and the universe of N_{η}^{∗} is A_{α_{η}}^{2}.

(d) If δ ∈ E, η ∈ ^{δ}2 and α_{η} = δ, then:

(d1) for i < f^{1}(δ) and η ◁ ν ∈ ^{δ+i+1}2, the model N_{ν}^{∗} is given by F applied to amalgamate M_{δ+i+1}^{1} and N_{ν↾(δ+i)}^{∗} over M_{δ+i}^{1}, using A_{δ+i+1}^{2} and

keeping a^{∗} out of M_{δ+i+1}^{∗}

(d2) for all ν, ν′ ∈ ^{δ+f^{1}(δ)+1}2 extending η if ν′ ≠ ν, \underline{then} for some β

tp(a^{∗}, M_{β}^{∗}, N_{ν}^{∗}) ≠ tp(a^{∗}, M_{β}^{∗}, N_{ν}^{∗})

(d3) for non-zero i ≤ f^{1}(δ) and ν such that η ◁ν ∈ ^{δ+i}2 we have M_{ν}^{∗} = M_{α_{ν}}^{1}

and α_{ν} = α_{ℓg(ν)} = α_{η} + i = α_{δ} + i (so M_{ν}^{∗} = M_{δ+i}^{1})

(d4) for non-zero i ≤ f^{1}(δ) and ν, ρ ∈ ^{δ+i}2 such that η ◁ ν & η ◁ ρ we have

N_{ν}^{∗} = N_{ρ}^{∗} call it N_{η}^{1}.

During carrying the definition the main point is guaranteeing clause (d). So let η ∈ ^{δ}2 and assume that α_{η} = δ ∈ E^{∗}. First we define by induction on i ≤ f^{1}(δ), a model N_{η,i}^{1} such that N_{i}^{1} has universe A_{δ+i}^{2}, N_{η,i}^{1} is ≤_{𝔎}-increasing, contradiction, M_{δ+i}^{1} ≤_{𝔎} N_{η,i}^{1} and N_{η,0}^{1} = N_{η}^{∗} and f, i < f^{1}(δ) then

N_{η,i+1}^{1} = F (M_{δ+i}^{1}, M_{δ+i+1}^{1}, N_{η,i}^{1}, A_{δ+i+1}^{2}, a^{∗}).

Next we choose by induction on i ≤ f^{1}(δ) for each ρ ∈ ^{i}2, an ordinal β_{η,ℓ} ∈ [δ + f^{1}(δ), λ^{+}) and model N_{η,ρ}^{1} such that:

(i) N_{η,ρ}^{1} ∈ K_{λ} has universe A_{β_{η,ρ}}^{2}

(ii) M_{β_{η,ρ}}^{1} ≤_{𝔎} N_{η,ρ}^{1} and (a^{∗}/ =^{N_{η,ρ}^{1}}) ∉ M_{β_{η,ρ}}^{1}/ =^{N_{η,ρ}^{1}}

(iii) β_{η,<>} = δ + f^{1}(δ)

(iv) ρ_{1} ◁ ρ_{2} ⇒ β_{η,ρ_{1}} < β_{η,ρ_{2}} & N_{η,ρ_{1}}^{1} ≤_{𝔎} N_{η,ρ_{2}}^{1}

(v) i limit ⇒ N_{η,ρ}^{1} = ⋃_{ζ<i} N_{η,ρ↾ζ}^{1}

(vi) β_{η,ρˆ <0>} = β_{ηˆ ρˆ <1>} and tp(a^{∗}, M_{β_{η,ρˆ <0>}}^{1}, N_{η,ρˆ <0>}^{1}) ≠ tp(a^{∗}, M_{β_{η,ρˆ <1>}}^{1}, N_{η,ρˆ <1>}^{1}).

There is no problem to carry the definition. Now let

α_{ηˆ ρ} = δ + i = ℓg(ηˆ ρ) if ρ ∈ ^{i}2 and i ≤ f^{1}(δ)

α_{ηˆ ρ} = β_{η,ρ} if ρ ∈ ^{f^{1}(δ)+1}2

M_{ηˆ ρ}^{∗} = M_{α_{ηˆ ρ}}^{1} if ρ ∈ ^{i}2, i ≤ f^{1}(δ) + 1

N_{ηˆ ρ}^{∗} = N_{η,i}^{1} if ρ ∈ ^{i}2 and i ≤ f^{1}(δ)

N_{ηˆ ρ}^{∗} = N_{η,ρ}^{1} if ρ ∈ ^{f^{1}(δ)+1}2.

Now check.

Having carried the induction for η ∈ ^{λ^{+}}2 we let M̄^{2,η} = ⟨N_{η↾α}^{∗} : α < λ^{+}⟩ and f^{2} = f^{1} + 1. We have to check that the demand in Definition 3.14 holds.

Note that this is essentially the proof mentioned in 3.16(9).

By our initial remarks there is little difficulty in carrying out this induction. We then set M̄^{2,η} = ⟨N_{η↾i}^{∗} : i < λ^{+}⟩ for η ∈ ^{λ^{+}}2. Given a set S ⊆ λ^{+} we consider M̄^{2,η} for η ∈ ^{λ^{+}}2 extending 0_{λ\S}, together with the function f^{2} equal to f^{1} + 1 on S and to f^{1} on S. We claim that the two conditions of Definition 3.21(1) are met.

The first of these is a condition on the type of construction allowed, and, of course, it has been obeyed, notably in (d1) above:

(∗)_{1} (M̄^{1}, f^{1}) ≤_{F,a}^{at} (M̄^{2,η}, f^{1}); and M^{2,η} ↾ α is determined by η ↾ α.

The second condition referred to a club E, which can be the intersection of the club we have defined above with {δ : α_{δ} = δ}. This condition goes as follows:

(∗)_{2} it is impossible to find sequences η^{3}, η^{4} (extending 0_{λ\S}), extensions (M̄^{1}, f^{2}) ≤_{F} (M̄^{3}, f^{3}), (M̄^{4}, f^{4}) witnessed by clubs E^{3}, E^{4} (i.e. E^{ℓ} is the intersection of the clubs which witness the atomic relations ≤_{F}^{at} implicit in ≤_{F}), and embeddings f_{ℓ} : M^{2,η^{ℓ}} → M^{ℓ} (ℓ = 3 or 4) over M^{1} such that for some δ ∈ E ∩ E^{3} ∩ E^{4} ∩ S we have:

(i) M̄^{3} ↾ (δ + f^{2}(δ) + 1) = M̄^{4} ↾ (δ + f^{2}(δ) + 1);

(ii) f^{3} ↾ δ = f^{4} ↾ δ;

(iii) η^{3} ↾ δ = η^{4} ↾ δ (call the restriction η) and η^{3}(δ) ≠ η^{4}(δ);

(iv) f_{3}, f_{4} are equal on M_{δ}^{2,η}; and

(v) for ℓ = 3, 4, f_{ℓ} maps M_{δ}^{2,η} into M_{δ}^{ℓ}.

Suppose on the contrary we have η^{3}, η^{4} (extending 0_{λ\S}),(M̄^{1}, f^{2}) ≤_{F} (M̄^{3}, f^{3}), (M̄^{4}, f^{4}), E^{3}, E^{4}, f_{3}, f_{4} and δ as above.

Let M̂ = M_{δ+f^{2}(δ)}^{3}. It follows from condition (i) and the fact that δ belongs to the witnessing clubs E^{3}, E^{4} that M̂ = M_{δ+f^{2}(δ)}^{4}. Then f_{3}, f_{4} provide embeddings of N_{η^{3}↾(δ+f^{1}(δ)+1)}^{∗} and N_{η^{3}↾(δ+f^{1}(δ)+1)}^{∗} into M̂ which agrees on N_{η}^{∗} (hence on a^{∗}) and on M_{η}^{1}. By 3.16(2) we are done. □_{3.25}

3.26 Lemma. Let 𝔎 be an abstract elementary class with LS(𝔎) ≤ λ which is categorical in λ and in λ^{+}, with 1 ≤ I(λ^{++}, K) < 2^{λ^{++}}. Assume that 2^{λ} < 2^{λ^{+}} < 2^{λ^{++}}, or at least that the definitional weak diamond holds for both λ^{+} and λ^{++}. \underline{Then}:

(∗) for any M ∈ K_{λ^{+}} and any triple (M^{0}, N^{0}, a^{0}) in K_{λ}^{3} with M^{0} ≤_{𝔎} M, we can find sequences M̄ = ⟨M_{i} : i < λ^{+}⟩, N̄ = ⟨N_{i} : i < λ^{+}⟩ such that

(a) (M^{0}, N^{0}, a^{0}) = (M_{0}, N_{0}, a^{0});

(b) (M_{i}, N_{i}, a) is increasing and continuous in K_{λ}^{3};

(c) the union of the M_{i} is M;

(d) the set S(M̄, N̄, a) is stationary in λ^{+}, where S(M̄, N̄, a) is the set of δ < λ^{+} such that for some j > δ for all i ≥ j

if we have (M_{j}, N_{j}, a) ≤_{ℓ} (M_{i}, N^{ℓ}, a) for ℓ = 1, 2 then we can amalgamate (M_{i}, N^{1}, a) and (M_{i}, N^{2}, a) over (M_{i}, N_{δ}, a).

Proof. Otherwise, we claim that any full λ^{+}-amalgamation choice function will have the λ^{+}-coding property.

Let M̄ ∈ Seq_{λ^{+}}, f : λ^{+} → λ^{+} and S ⊆ λ^{+} be given. Then as K is categorical in λ^{+}, we may suppose that M is the union of the M_{i}. If (c) fails, then (d) fails on a club of δ, providing enough failures of amalgamation to carry the proof as in 3.12. □_{3.12}

3.27 Lemma. Let 𝔎 be an abstract elementary class with LS(𝔎) ≤ λ which is categorical in λ, λ^{+} and λ^{++} and with no model in cardinality λ^{+3}.

Suppose that there is no model in K_{λ^{+}} saturated above λ and that:

(∗′) for any M ∈ K_{λ^{+}} and any triple (M^{0}, N^{0}, a^{0}) in K_{λ}^{3} with M^{0} ≤_{𝔎} M, we can find sequences M̄ = ⟨M_{i} : i < λ^{+}⟩, N̄ = ⟨N_{i} : i < λ^{+}⟩ such that

(a) (M^{0}, N^{0}, a^{0}) = (M_{0}, N_{0}, a^{0});

(b) (M_{i}, N_{i}, a) is increasing and continuous in K_{λ}^{3};

(c) the union of the M_{i} is M;

(d) the set S(M̄, N̄, a) is stationary in λ^{+} where S(M̄, N̄, a) is the set of δ < λ^{+} such that for some j > δ for all i ≥ j

if we have (M_{j}, N_{j}, a) ≤_{h_{ℓ}} (M_{i}, N^{ℓ}, a) for ℓ = 1, 2 then we can amalgamate (M_{i}, N^{1}, a) and (M_{i}, N^{2}, a) over (M_{i}, N_{δ}, a).

\underline{Then} the minimal triples are dense in K_{λ}^{3}.

Proof. Suppose that there is no minimal triple above (M^{∗}, N^{∗}, a^{∗}). It suffices to show that there is no maximal model in K_{λ^{++}} and as K_{λ^{++}} is categorical, this will follow from the existence of a single pair of models (M′, N′) in K_{λ^{++}} with M′ <_{𝔎} N′. So it suffices to show:

every triple (M, N, a) in K_{λ^{+}}^{3} has a proper extension in K_{λ^{+}}^{3}

as the desired pair (M′, N′) can then be built as the limit of an increasing continuous chain.

Fix (M, N, a) in K_{λ^{+}}^{3}. As there is no model saturated over λ in K_{λ^{+}}, there is some M_{0} in K_{λ} over which there are more than λ^{+} types. By λ-categoricity we may suppose M_{0} ≤_{𝔎} M. Fix a triple (M_{0}, N_{0}, b) in K_{λ}^{3} for which tp(b, M_{0}, N_{0}) is not realized in M.

Apply (∗′) to M and (M_{0}, N_{0}, b) to get sequences M̄, N̄ of length λ^{+} as in (∗′). Let S = S(M̄, N̄, b). In particular M = ⋃ M_{i}. Let M^{1} = ⋃ N_{i}. As 𝔎 has amalgamation in λ^{+} we may suppose M^{1}, N ≤_{𝔎} N^{1} with N^{1} ∈ 𝔎_{λ^{+}}.

We can also choose an increasing continuous sequence (M′_{i}, N′_{i}, a^{∗}) beginning with (M^{∗}, N^{∗}, a^{∗}) such that each (M′_{i}, N′_{i}, a^{∗}) is reduced and tp(a^{∗}, M′_{i}, N′_{i}) has more than one extension in 𝒮 (M′_{i+1}), using the failure of minimality and 2.7(1). By categoricity we may suppose M = ⋃ M′_{i}. Set M^{2} = ⋃ N′_{i}. By amalgamation we may suppose M^{2}, N^{1} ≤_{𝔎} N^{2} ∈ K_{λ^{+}}.

We claim that one of the triples (M^{1}, N^{1}, a) or (M^{2}, N^{2}, a) is a proper extension of (M, N, a). Suppose on the contrary that a belongs to both M^{1} and M^{2}.

Represent N^{2} as the union of a continuous ≤_{𝔎}-increasing chain ⟨N_{i}^{∗} : i < λ^{+}⟩ of models in K_{λ}. Let E be

{i < λ^{+} : M_{i} = M′_{i}; N_{i}^{∗} ∩ M = M_{i}; N_{i}^{∗} ∩ M^{1} = N_{i}; N_{i}^{∗} ∩ M^{2} = N′_{i}},

a club in λ^{+}.

Fix δ ∈ E ∩ S such that a is in N_{δ} and N′_{δ}. We show now that a^{∗} ∈ N_{δ}. Now (M′_{δ}, N′_{δ}, a^{∗}) is reduced. If a^{∗} ∉ N_{δ} then (N_{δ}, N^{2}, a^{∗}) lies over (M′_{δ}, N′_{δ}, a^{∗}) and hence N_{δ} ∩ N′_{δ} ⊆ M′_{δ}; but the element a witnesses the failure of this condition. So a^{∗} ∈ N_{δ}.

Let j > δ be chosen in accordance with the definition of S(M̄, N̄, b) and let i > j′ > j_{a}, k, j′ ∈ E. As tp(a^{∗}, M′_{j′}, N′_{j′}) has more than one extension to M′_{j′+1}, the same applies to M′_{i}. However, M′_{i} = M_{i} and M′_{j′} = M_{j′} and thus tp(a^{∗}, M_{j′}, N_{j′}^{2}) = tp(a^{∗}, M_{j′}, N_{j′}^{∗}) = tp(a^{∗}, M_{j′}, N_{j′}) has more than one extension over M_{i}. Thus, M_{i} and N_{j′} may be amalgamated in two incompatible ways over M_{j′}, getting N^{+} and N^{−}, say (i.e. the N^{+} and N^{−} cannot be amalgamated over M_{i} preserving the images of a^{∗}). Furthermore (M_{i}, N^{±}, b) lies above (M_{j}, N_{j}, b) in K_{λ^{+}}^{3}, that is, b is not mapped into M_{i}, because M does not realize tp(b, M_{0}, N_{0}). However, this contradicts the definition of S, as the triples (M_{i}, N^{+}, b) and (M_{i}, N^{−}, b); cannot be amalgamated over (M_{i}, N_{δ}) since a^{∗} belongs to N_{δ}. □_{3.27} 3.28 Theorem. Let 𝔎 be an abstract elementary class with LS(𝔎) ≤ λ which is categorical in λ and in λ^{+} with 1 ≤ I(λ^{++}, K) < 2^{λ^{++}}. Assume that 2^{λ} < 2^{λ^{+}} < 2^{λ^{++}}, or at least that the definitional weak diamond holds for both λ^{+} and λ^{++}.

Then under either of the following assumptions, the minimal triples are dense in K_{λ}^{3}:

(A) K is categorical in λ^{++} and has no model in cardinality λ^{+3}; (B) there is a model saturated above λ in cardinality λ^{+}.

Proof. By the previous lemmas 3.25, 3.26, 3.27. Note that (∗′) is exactly the negation of (∗). □_{3.28}

3.29 Remark. This will be proved without the additional assumptions (A, B) in [Sh 600]. In any case this does not affect the proof of Theorem 0.4

3.30 Claim. Let 𝔎 be an abstract elementary class with LS(𝔎) ≤ λ which is categorical in λ and in λ^{+}, with 1 ≤ I(λ^{++}, K) < 2^{λ^{++}}, and with no model in cardinality λ^{+3}. Assume that 2^{λ} < 2^{λ^{+}} < 2^{λ^{++}}.

\underline{Then} the minimal triples are dense in K_{λ}^{3}.

Proof. If 2^{λ^{+}} > λ^{++} we get the conclusion by 2.7. If 2^{λ^{+}} = λ^{++} then as 2^{λ} < 2^{λ^{+}} we have 2^{λ} = λ^{+}. Thus, there is a model in K_{λ^{+}} which is saturated above λ, and Lemma 3.28 applies. □_{3.30}

§4 Minimal types

We return to the analysis of minimal types initiated in 2.12. We use from §2 only 2.1, 2.6, 2.9 so there are repetitions.

4.1 Hypothesis.

(a) 𝔎 is an abstract elementary class with LS(𝔎) ≤ λ (for simplicity K_{<λ} = ∅).

(b) 𝔎 is categorical in λ, λ^{+} with K_{λ^{+2}} ≠ ∅ (note: (a) + (b) = (∗)_{λ}^{3} of 2.4).

(c) 𝔎 has amalgamation in λ (2.2(1)), so by (a) + (c), we have (∗)_{λ}^{2} from 2.9 hence 𝔎 satisfies the model theoretic properties which were deduced in 2.4-2.6,2.9 in particular:

(i) every (M, N, a) ∈ K_{λ}^{3} has the weak extension property 2.4; (ii) criteria for the extension property 2.9; (iii) basic Definitions and properties 2.3, 2.6.

4.2 Definition. 1) If p ∈ 𝒮 (N), N ∈ K_{λ} and N′ ∈ K_{λ} remember (from Definition 2.13)

𝒮_{p}(N′) = {f(p) : f is an isomorphism from N onto N′}

and let

𝒮_{≥p}(N′) = {q ∈ 𝒮 (N′) : q not algebraic (i.e. not realized by any
c ∈ N′) and, for some N′′ ∈ K_{λ},
N′′ ≤_{𝔎} N′, we have q ↾ N′′ ∈ 𝒮_{p}(N′′)}.

2) We say the type p ∈ 𝒮 (N) is λ-algebraic if ∥N∥ ≤ λ and for every M such that N ≤_{𝔎} M we have: λ ≥ |{c ∈ M : tp(c, N, M) = p}|.

4.3 Claim. If (M_{0}, M_{1}, a) ∈ K_{λ}^{3} is minimal, \underline{then} it has the extension property.

Proof. Let p^{∗} = tp(a, M_{0}, M_{1}), and assume it is a counter-example. We note:

⨂1 for some M^{∗} we have M_{0} ≤_{𝔎} M^{∗} ∈ K_{λ} but for no M^{+} and b do we have M^{∗} ≤_{𝔎} M^{+} ∈ K_{λ}, b ∈ M^{+}\M^{∗} and b realizes p^{∗}. [Why? If not for every N, M_{0} ≤_{𝔎} N ∈ K_{λ}, we can find N_{1}, N ≤_{𝔎} N_{1} ∈ K_{λ} and b ∈ N_{1}\N which realizes p^{∗}. Hence (as 𝔎_{λ} has amalgamation in λ) we can find N_{2} such that N_{1} ≤_{𝔎} N_{2} ∈ K_{λ}, and g a ≤_{𝔎}-embedding of M_{1} into N_{2} extending id_{M_{0}} such that g(a) = b. This proves the extension property].

⨂2 if p ∈ 𝒮_{≥p^{∗}}(N) and N ∈ K_{λ} and N ≤_{𝔎} N^{∗} ∈ K, \underline{then} the set of elements of b ∈ N^{∗} realizing p has cardinality ≤ λ. [Why? by 4.1(c)(ii); so indirectly 2.9(2)].

⨂3 if N ∈ K_{λ}, then |𝒮_{≥p^{∗}}(N)| > λ^{+}.

Proof of ⨂_{3}. If N forms a counterexample, as K is categorical in λ and using ⊗_{2} we can find ⟨N_{i} : i < λ^{+}⟩, ≤_{𝔎}-increasing continuous sequence of members of K_{λ} such that:

(∗) for every α < λ^{+} and q ∈ 𝒮_{≥p^{∗}}(N_{α}), for some β = β_{q} < λ^{+} we have: for no N′, b do we have N_{β} ≤_{𝔎} N′ ∈ K_{λ}, b ∈ N′\N_{β} and b realizes q.

So N_{λ^{+}} = ⋃_{i<λ^{+}} N_{i} has the property

(∗∗) if N̄′ = ⟨N′_{α} : α < λ^{+}⟩ is a representation of N_{λ^{+}} then for a club of δ < λ^{+} for every q ∈ 𝒮_{≥p^{∗}}(N′_{δ}) for a club of β ∈ (δ, λ^{+}), for no N′, b do we have:

N_{β} ≤_{𝔎} N′ ∈ K_{λ}, b ∈ N′\N_{β} and b realizes q.

On the other hand, we can choose by induction on α < λ^{+} a triple (N_{0,α}, N_{1,α}, a) ∈ K_{λ}^{3} increasing continuous in α such that (N_{00}, N_{10}, α) = (M_{0}, M_{1}, a) and N_{0,α} ≠ N_{0,α+1} (existence by the weak extension property; i.e. 2.4 = 4.1(c)(i)). Now N_{0,λ^{+}} = ⋃_{α<λ^{+}} N_{0,α} ∈ K_{λ^{+}} does not satisfy the statement (∗∗):

⟨N_{0,α} : α < λ^{+}⟩ is a representation of N_{0,λ^{+}}, and for every α, tp(a, N_{0,α}, N_{1,α}) extend tp(a, N_{0,0}, N_{1,0}) = tp(a, M_{0}, M_{1}) = p^{∗} hence tp(a, N_{0,α}, N_{1,α}) ∈ 𝒮_{≥p^{∗}}(N_{0,α}) satisfies: for every β ∈ (α, λ^{+}), there is N′, N_{0,β} ≤_{𝔎} N′ and some b ∈ N′\N_{0,β} realizes tp(a, N_{0,α}, N_{1,α}); simply choose (N′, b) = (N_{1,β}, a). So N_{0,λ^{+}}, N_{λ^{+}} cannot be isomorphic (as one satisfies (∗∗) the other not). But both are in K_{λ^{+}}, contradicting the categoricity of 𝔎 in λ^{+}. □_{⊗_{3}}

To finish the proof of 4.3 it is enough to prove

4.4 Claim. If p^{∗} ∈ 𝒮 (M_{0}) is minimal, M_{0} ∈ K_{λ}, \underline{then} N ∈ K_{λ} ⇒ |𝒮_{≥p^{∗}}(N)| ≤ λ^{+}.

Proof. By 4.6, 4.7 below. Note that 𝒮_{≥p^{∗}}(N) has the same cardinality for every N ∈ K_{λ}.

4.5 Claim. 1) If N_{1} ≤_{𝔎} N_{2} are in K_{λ} and p_{1} ∈ 𝒮 (N_{1}) is minimal and is omitted by N_{2} \underline{then} p_{1} has a unique extension in 𝒮 (N_{2}), call it p_{2}, and p_{1} ∈ 𝒮_{≥p^{∗}}(N_{1}) ⇒ [p_{2} ∈ 𝒮_{≥p^{∗}}(N_{2}) and p_{2} is minimal].

2) If N_{1} ≤_{𝔎} N_{2} are in K_{λ}, p_{1} ∈ 𝒮 (N_{1}) minimal, \underline{then} p_{1} has at most one nonalgebraic extension in 𝒮 (N_{2}) called p_{2}, it is minimal and p_{1} ∈ 𝒮_{≥p^{∗}}(N_{1}) ⇒ p_{2} ∈ 𝒮_{≥p^{∗}}(N_{2}).

3) (Continuity) if ⟨N_{i} : i ≤ α⟩ is a ≤_{𝔎}-increasing continuous sequence of members of K_{λ}, p_{i} ∈ 𝒮 (N_{i}), p_{0} minimal, p_{i} ∈ 𝒮 (N_{i}) extends p_{0} and is non-algebraic then ⟨p_{i} : i ≤ α⟩ is increasing continuously.

Proof of 4.5. Easy. E.g.,

3) If i < j ≤ α then p_{j} ↾ N_{i} is well defined, it belongs to 𝒮 (N_{i}), also it is nonalgebraic and extending p_{0} hence by the uniqueness (=4.5(1)) we have p_{i} = p_{j} ↾ N_{i}. If δ ≤ α, p_{δ} ∈ 𝒮 (N_{δ}) extends p_{i} for i < δ; if p′_{δ} ∈ 𝒮 (N_{δ}) extends each p_{i}(i < δ) then it extends p_{0} and is non-algebraic hence by uniqueness p′_{δ} = p_{δ}.

4.6 Claim. If N ∈ K_{λ}, 𝒮 ⊆ 𝒮 (N) and |𝒮 | > λ^{+}, \underline{then} we can find N^{∗}, N_{i} in K_{λ}, (for i < λ^{++}) such that:

(α) N ≤_{𝔎} N^{∗} <_{𝔎} N_{i} (β) for no i_{0} < i_{1} < λ^{++} and c_{ℓ} ∈ N_{i_{ℓ}}\N^{∗} (for ℓ = 0, 1) do we have

tp(c_{0}, N^{∗}, N_{i_{0}}) = tp(c_{1}, N^{∗}, N_{i_{1}}) (γ) there are a_{i} ∈ N_{i} (for i < λ^{++}) such that tp(a_{i}, N, N_{i}) ∈ 𝒮 (and they are

pairwise distinct).

Remark. We use here less than Hypothesis 4.1:

(∗) 𝔎 is abstract elementary class with amalgamation in λ, categorical in λ, K_{λ^{+}} ≠ ∅.

The same applies to 4.7.

Proof. Without loss of generality |N| = λ; now choose by induction on α < λ^{++}, N̄^{α}, N_{α}, a_{α} such that:

(A) N_{α} ∈ K_{λ^{+}} has a set of elements λ × (1 + α) and N_{α} is ≤_{𝔎}-increasing

continuous in α (B) N̄^{α} = ⟨N_{i}^{α} : i < λ^{+}⟩ is a representation of N_{α} (i.e. is ≤_{𝔎}-increasing

continuous, ∥N_{i}^{α}∥ ≤ λ and N_{α} = ⋃_{i<λ^{+}} N_{i}^{α})

(C) for α < λ^{++} successor, if i < j < λ^{+}, p ∈ 𝒮 (N_{i}^{α}) is realized in N_{j}^{α} and is λ-algebraic (see Definition 4.2(2)) \underline{then} for no N′, b do we have N_{j}^{α} ≤_{𝔎} N′ ∈ K_{λ} and b ∈ N′\N_{j}^{α} realizes p (actually not needed)

(D) N = N_{0}^{0} ≤_{𝔎} N_{0} and a_{α} ∈ N_{α+1}\N_{α} realizes some p_{α} ∈ 𝒮 not realized in

N_{α}. (E) If ℵ_{0} < cf(α) ≤ λ \underline{then} let for j < λ:

M_{j}^{α} = ∩{ ⋃_{β∈C} N_{j}^{β} : C a club of α} whenever possible, i.e. the result is in

K_{λ} and ≤_{𝔎} N^{α} (F) for each α < λ^{++}, for a club E_{α}^{0} of ordinals i < λ^{+} we have (N_{i}^{α}, N_{i}^{α+1}, a_{α})

is reduced;

hence (as tp(a_{α}, N_{i}^{α}, N_{i}^{α+1}) extends tp(a_{α}, N, N_{i}^{α+1}) which is not realized in N_{α}):

(G) for every (i ∈ E_{α}^{0} and) b ∈ N_{i}^{α+1}\N_{i}^{α} the type tp(b, N_{i}^{α}, N_{i}^{α+1}) is not realized in N_{α} (a key point).

There is no problem to carry out the construction (𝔎 has amalgamation in λ). Let w_{i}^{α} =: {β : N_{i}^{α+1} ∩ N_{β+1} ⊈ N_{β}}, so necessarily |w_{i}^{α}| ≤ λ, w_{i}^{α} is increasing continuous in i < λ^{+} and α = ⋃_{i<λ^{+}} w_{i}^{α} and for β < α let

i(β, α) = Min{i : β ∈ w_{i}^{α}}. Now for every α ∈ S^{∗} =: {δ < λ^{++} : cf(δ) = λ^{+}}, the set

E_{α} =: {i < λ^{+} : i limit , N ≤_{𝔎} N_{i}^{α+1}, a_{α} ∈ N_{i}^{α+1},

and for every β < α if

β ∈ w_{i}^{α} then

N_{i}^{β} = N_{i}^{α} ∩ N_{β} and for j < i the closure of w_{j}^{α} (in α) is included in w_{i}^{α}

and β_{1} < β_{2} & β_{1} ∈ w_{j}^{α} & β_{2} ∈ w_{j}^{α} ⇒ i(β_{1}, β_{2}) < i}

is a club of λ^{+}.

As we can assume λ > ℵ_{0} (ignoring λ = ℵ_{0} as was treated earlier in [Sh 88] though for a PC_{ℵ_{0}} class, or see [Sh 600],§2), we can choose j_{α} ∈ E_{α} such that cf(j_{α}) = ℵ_{1} and let δ_{α} = sup(w_{j_{α}}^{α}), now w_{j_{α}}^{α} is closed under ω-limits (as ⟨w_{j}^{α} : j ≤ α⟩ is increasing continuous, j < α ⇒ closure(w_{j}^{α}) ⊆ w_{j+1}^{α}) and ℵ_{1} = cf(otp w_{j_{α}}^{α}) so there is ⟨β_{ε} : ε < ω_{1}⟩ increasing continuous with limit δ_{α}, β_{ε} ∈ w_{j_{α}}^{α} so ε < ζ < ω_{1} ⇒ N_{j}^{β_{ε}} = N_{j}^{β_{ζ}} ∩ N_{β_{ε}} and easily N_{j}^{α} ∩ N_{β_{ε}} = N_{j}^{β_{ε}} hence

⨁ M_{j_{α}}^{δ_{α}} = ∩{ ⋃_{j∈C} N_{j}^{β} : C a club of δ_{α}}

so N_{j_{α}}^{α} = M_{j_{α}}^{δ_{α}} (see [Sh 351, §4]).

By Fodor lemma for some j^{∗}, α^{∗} and stationary S ⊆ S^{∗}, α ∈ S^{∗} ⇒ j_{α} = j^{∗} & δ_{α} = δ^{∗}. So for all α ∈ S, N_{j_{α}}^{α} are the same say N^{∗}. So N^{∗} ∈ K_{λ}, for α ∈ S, q_{α} = tp(a_{α}, N^{∗}, N_{j_{α}}^{α+1}) extend p_{α}(∈ 𝒮). Also if r ∈ 𝒮 (N^{∗}) is realized in N_{j_{α}}^{α+1} say by b (for some α ∈ S) then no member of ⋃{N_{j^{∗}}^{β+1}\N_{j^{∗}}^{β} : β ∈ S∩α} realizes it (holds by clause (G), see clause (F)). So the sets Γ_{α} = {tp(b, N_{j^{∗}}^{δ^{∗}}, N_{j^{∗}}^{α+1}) : b ∈ N_{j^{∗}}^{α+1}\N^{∗}} are pairwise disjoint and each has a member extending p_{α} ∈ 𝒮 (as exemplified by a_{α} and p_{α} is not extended by any p ∈ ⋃_{β<α} Γ_{β} (as p_{α} is not realized in N_{α})). □_{4.6}

4.7 Claim. Assume p^{∗} is a counterexample to 4.4.

1) If N ∈ K_{λ}, Γ ⊆ 𝒮_{≥p^{∗}}(N), |Γ| ≤ λ^{+} \underline{then}

{p ∈ 𝒮_{≥p^{∗}}(N) : for some N′, N ≤_{𝔎} N′ ∈ K_{λ} and some b ∈ N′

realizes p but no b ∈ N′ realizes any q ∈ Γ}

has cardinality ≥ λ^{++}.

2) We can find N ∈ K_{λ^{+}}, and N_{i}, N <_{𝔎} N_{i} ∈ K_{λ^{+}} for i < 2^{λ^{+}} such that the set Γ_{i} = {tp(a, N, N_{i}) : a ∈ N_{i}} are pairwise distinct, in fact, no one embeddable into another (so we get I(λ^{+}, 𝔎) = 2^{λ^{+}} and if (2^{λ})^{+} < 2^{λ^{+}} then IE(λ^{+}, 𝔎) = 2^{λ^{+}} thus contradicting the categoricity in λ^{+} from the assumptions).

Proof. 1) Apply 4.6 with N and 𝒮_{≥p^{∗}}(N) here standing for N and 𝒮 there, so we get N^{∗}, N_{i}(i < λ^{++}) and M^{∗} such that N^{∗} ≤_{𝔎} N_{i} ∈ 𝔎_{λ}, and Γ_{i} = {tp(a, N^{∗}, N_{i}) : a ∈ N_{i}\N^{∗}} are pairwise disjoint and there are p_{i} ∈ Γ_{i}, p_{i} ↾ N ∈ 𝒮_{≥p^{∗}}(N) pairwise distinct; now p_{i} is not algebraic hence p_{i} ∈ 𝒮_{≥p^{∗}}(N^{∗}). As K is categorical in λ, without loss of generality N^{∗} = N, so all but ≤ λ^{+} of the models N_{i} can serve as the required N′.

2) Now by part (1) of 4.7 we can choose by induction on i < λ^{+}, ⟨(N_{η}, Γ_{η}) : η ∈ ^{i}2⟩ such that:

(a) N_{η} ∈ K_{λ} and Γ_{η} ⊆ ⋃_{j<i} 𝒮_{≥p^{∗}}(N_{η↾j}) and |Γ_{η}| ≤ λ

(b) if ν ◁ η then N_{ν} ≤_{𝔎} N_{η} and Γ_{η} ⊆ Γ_{ν}

(c) some p ∈ Γ_{ηˆ ⟨0⟩} is from 𝒮 (N_{η}) and is realized in N_{ηˆ ⟨1⟩} (similarly for Γ_{ηˆ ⟨1⟩}, N_{ηˆ ⟨0⟩})

(d) if i is a limit ordinal, then N_{η} = ⋃_{j<i} N_{η↾j} and Γ_{η} = ⋃_{j<i} Γ_{η↾j}.

The successor case is done by 4.7(1) (you may object that the type in Γ_{η} are not from 𝒮_{≥p^{∗}}(N_{η}) but from ⋃_{j<i} 𝒮_{≥p^{∗}}(N_{η↾j})? However they are minimal - see 4.5(1)).

For η ∈ ^{λ^{+}}2 let N_{η} = ⋃_{i<λ^{+}} N_{η↾i}. Now by 1.4(1), {N_{η}/ ≅: η ∈ ^{λ^{+}}2} has cardinality

2^{λ^{+}}, contradiction. □_{4.7}, □_{4.4}, □_{4.3}

4.8 Claim. 1) If M_{0} ∈ K_{λ}, M_{1} ∈ K_{λ^{+}}, and M_{0} ≤_{𝔎} M_{1} \underline{then} every minimal p ∈ 𝒮 (M_{0}) is realized in M.

2) Every M_{1} ∈ K_{λ^{+}} is saturated at least for minimal types (i.e. if M_{0} ≤_{𝔎} M_{1}, M_{0} ∈ K_{λ} and M_{1} ∈ K_{λ^{+}} \underline{then} every minimal p ∈ 𝒮 (M_{0}) is realized in M_{1}).

3) If M ∈ K_{λ} then {p ∈ 𝒮 (M) : p minimal} has cardinality ≤ λ^{+}.

Proof. 1) Let N̄ = ⟨N_{α} : α < λ^{+}⟩ be a representation of N_{λ^{+}} ∈ K_{λ^{+}}. Let N ∈ K_{λ}, p ∈ 𝒮 (N) be minimal we ask

(∗)_{p} is there a club of α < λ^{+} such that every q ∈ 𝒮_{≥p}(N_{α}) is realized in λ^{+}?

By 4.4 there is N′_{λ^{+}} ∈ K_{λ^{+}} for which the answer is yes, hence, as 𝔎 is categorical in λ^{+}, this holds for N_{λ^{+}}. So this holds for every minimal p. Now if N′ ≤_{𝔎} N_{λ^{+}}, N′ ∈ K_{λ} and p ∈ 𝒮 (N′) is minimal then for some α, N′ ≤_{𝔎} N_{α} and for every β ∈ [α, λ^{+}), p has a unique non-algebraic extension p_{β} ∈ 𝒮 (N_{β}) (which necessarily is minimal, exist by 4.3). Now p_{β} ∈ 𝒮_{≥p}(N_{β}) hence for a club of β < λ^{+}, p_{β} is realized in N_{λ^{+}}, so we have finished the proof of part (1).

2) By part (1).

3) Follows by part (1). □_{4.8}

From 4.3, 2.9(1) we can conclude

4.9 Conclusion. Every (M_{0}, M_{1}, a) ∈ K_{λ}^{3} has the extension property.

§5 Inevitable types and stability in λ

5.1 Hypothesis. Assume the model theoretic assumptions from 4.1 and

(d) there is a minimal member of K_{λ}^{3} (follows from the conclusion of ?).

5.2 Definition. We call p ∈ 𝒮 (N) inevitable if: N ≤_{𝔎} M & N ≠ M ⇒ some c ∈ M realizes p. We call (M, N, a) ∈ K_{λ}^{3} inevitable if tp(a, M, N) is inevitable.

Now using 4.3 - 4.8 we shall deduce

5.3 Claim. 1) If there is a minimal triple in K_{λ}^{3}, \underline{then} there is an inevitable p = tp(a, N, N_{1}) with (N, N_{1}, a) ∈ K_{λ}^{3} minimal.

2) Moreover, if p_{0} ∈ 𝒮 (N_{0}) is minimal, N_{0} ∈ K_{λ} \underline{then} we can find N_{1}, N_{0} ≤_{𝔎} N_{1} ∈ K_{λ} such that the unique non-algebraic extension p_{1} of p_{0} in 𝒮 (N_{1}) is inevitable.

Proof of 5.3. 1) Follows by part (2).

2) Let (M_{0}, M_{1}, a) ∈ K_{λ}^{3} be minimal and p_{0} = tp(a, M_{0}, M_{1}). We try to choose by induction on i a model N_{i} such that: N_{0} = M_{0}, N_{i} ∈ K_{λ} is ≤_{𝔎}-increasing continuously and N_{i} omits p_{0}, N_{i} ≠ N_{i+1}. If we succeed, ⋃_{i<λ^{+}} N_{i} is a member of

K_{λ^{+}} which is non-saturated for minimal types, contradicting 4.8(2). As for i = 0, i limit we can define, necessarily for some i we have N_{i} but not N_{i+1}. Now p_{0} has a unique extension in 𝒮 (N_{i}) which we call p_{i} and p_{0} has no algebraic extension in 𝒮 (N_{i}). [Why? As N_{i} omits p_{0}]. So p_{i} is the unique extension of p_{0} in 𝒮 (N_{i}) [by 4.5(1)], and so

(∗) if N_{i} ≤_{𝔎} N′ ∈ K_{λ} and N′ ≠ N, \underline{then} p_{i} is realized in N′.

By L.S. we can omit “N′ ∈ K_{λ}”, so (N_{i}, p_{i}) are as required. □_{5.3}

5.4 Fact. Inevitable types have few (≤ λ) conjugates (i.e. for p ∈ 𝒮 (M_{0}) inevitable M_{0} ∈ K_{λ}, M_{1} ∈ K_{λ} we have |𝒮_{p}(M_{1})| ≤ λ), moreover |{p ∈ 𝒮 (N) : p inevitable}| ≤ λ for N ∈ K_{λ}.

Proof. Easy.

The following construction shall play a central role in this paper.

5.5 Claim. For any limit α < λ^{+}, we can find ⟨N_{i} : i ≤ α⟩ and ⟨p_{i} : i ≤ α⟩ such that:

(i) N_{i} ∈ K_{λ}, (ii) N_{i} is ≤_{𝔎}-increasing continuous (iii) p_{i} ∈ 𝒮 (N_{i}) is minimal, (iv) p_{i} increases continuously (see 4.5(3))

(v) p_{0} is inevitable

(vi) p_{α} is inevitable

(vii) N_{i} ≠ N_{i+1} moreover some c ∈ N_{i+1}\N_{i} realizes p_{0} (hence p_{i}).

Remark. Why not just build a non-saturated model in order to prove 5.5? Works, too.

Proof. Choose N^{0} <_{𝔎} N^{1} in K_{λ^{+}} (so N^{0} ≠ N^{1}), such a pair exists as K_{λ^{+2}} ≠ ∅. Let N^{ℓ} = ⋃_{i<λ^{+}} N_{i}^{ℓ} with N_{i}^{ℓ} ∈ K_{λ} being ≤_{𝔎}-increasing continuously in i. Now

E_{0} = {δ < λ^{+} : N_{δ}^{1} ≠ N_{δ}^{0} and N_{δ}^{1} ∩ N^{0} = N_{δ}^{0}} is a club of λ^{+}. Without loss of generality E_{0} = λ^{+}.

For each c ∈ N^{1}\N^{0}, the set

X_{c} =: {i < λ^{+} : c ∈ N_{i}^{1} and (N_{i}^{0}, N_{i}^{1}, c) is minimal}

is empty or an end segment of λ^{+} hence

E_{1} = {δ < λ^{+} :(i) δ limit and N_{δ}^{1} ⊈ N^{0}
(ii) if i < δ and p ∈ 𝒮 (N_{i}^{0}) is minimal inevitable
and realized in N^{0}\N_{δ}^{0} then it is
realized in N_{δ}^{0}\N_{i}^{0} (actually automatic)
(iii) if c ∈ N_{δ}^{1}\N^{0} (hence ∃i < δ, c ∈ N_{i}^{1}) and X_{c}
is non-empty \underline{then} δ ∈ X_{c} and min(X_{c}) < δ\}

is a club of λ^{+} (see 5.4).

Now for δ ∈ E_{1}, we have N_{δ}^{0} <_{𝔎} N_{δ}^{1}, so by 5.3(1) there is c_{δ} ∈ N_{δ}^{1}\N_{δ}^{0} such that:

(N_{δ}^{0}, N_{δ}^{1}, c_{δ}) is minimal
tp(c_{δ}, N_{δ}^{0}, N_{δ}^{1}) is inevitable

As δ is limit, for some i < δ, c ∈ N_{i}^{1}, also δ ∈ X_{c} hence there is j such that: i < j < δ & j ∈ X_{c} hence (N_{j}^{0}, N_{j}^{1}, c) is minimal; choose such j_{δ}, c_{δ}. Let κ = cf(κ) = cf(α) ≤ λ, so for some j^{∗}, c^{∗} we have

S = {δ ∈ E_{1} : cf(δ) = κ, j_{δ} = j^{∗}, c_{δ} = c^{∗}}

is stationary in λ^{+}.

Choose e closed ⊆ E_{1} of order type α + 1 with first element and last element in S; for ζ ∈ [j^{∗}, λ^{+}) let p_{ζ} = tp(c^{∗}, N_{ζ}^{0}, N_{ζ}^{1}). (In fact, we could have: all nonaccumulation member of e are in S; no real help.)

Now ⟨N_{ζ}^{0}, p_{ζ} : ζ ∈ e⟩ is as required (up to re-indexing)(clause (viii) holds by clause (ii) in the definition of E_{1}). □_{5.5} 5.6 Claim. Assume ⟨N_{i}, p_{i} : i ≤ α⟩ is as in 5.5, α < λ divisible by λ. \underline{Then} any p ∈ 𝒮 (N_{0}) is realized in N_{α}, moreover N_{α} is universal in K_{λ} over N_{0}.

Proof. (Similar to the proof of 0.24; which is [Sh 300, II,§3]).

Let N_{0} ≤_{𝔎} M_{0} ∈ K_{λ}, a ∈ M_{0}\N_{0} we shall show that tp(a, N_{0}, M_{0}) is realized in N_{α}.

Let α = ⋃_{i<λ} S_{i}, ⟨S_{i} : i < λ⟩ pairwise disjoint, each S_{i} unbounded in α, λ divides

otp(S_{i}) and Min(S_{i}) ≥ i. We choose by induction on i ≤ α the following:

N_{i}^{1}, M_{i}^{1}, h_{i}, ⟨a_{ζ} : ζ ∈ S_{i}⟩ (the last one only if i < α)

such that:

(a) N_{i}^{1} ≤_{𝔎} M_{i}^{1} are in K_{λ}

(b) N_{i}^{1} is ≤_{𝔎}-increasing continuous in i

(c) M_{i}^{1} is ≤_{𝔎}-increasing continuous in i

(d) (N_{0}^{1}, M_{0}^{1}) = (N_{0}, M_{0})

(e) ⟨a_{ζ} : ζ ∈ S_{i}⟩ is a list of {c ∈ M_{i}^{1} : c realizes p_{0}}

(f) h_{i} is an isomorphism from N_{i} onto N_{i}^{1}

(g) j < i ⇒ h_{j} ⊆ h_{i} and h_{0} = id_{N_{0}}

(h) a_{i} ∈ N_{i+1}^{1} (note: M_{i}^{1} ∩ N_{i+1}^{1} ≠ N_{i}^{1} in general).

\underline{For i = 0}: See clauses (d), (g)

N_{0}^{1} = N_{0}, M_{0}^{1} = M_{0}, h_{0} = id_{N_{0}}.

\underline{For i = limit}: Let N_{i}^{1} = ⋃_{j<i} N_{j}^{1} and M_{i}^{1} = ⋃_{j<i} M_{j}^{1} and h_{i} = ⋃_{j<i} h_{j} and lastly choose

⟨a_{ζ} : ζ ∈ S_{i}⟩ by clause (e).

\underline{For i = j + 1}: Note a_{j} is already defined, it belongs to M_{j}^{1} and it realizes p_{0}.

\underline{Case 1}: a_{j} ∈ N_{j}^{1} (so clause (h) is no problem). Use amalgamation on N_{j}, N_{i}, M_{j}^{1} and the mapping id_{N_{j}}, h_{i},i.e.

N_{i} ⟶ M_{i}^{1}
id_{N_{j}} ↑  ↑^{id_{N^{1}_{0}}}
N_{j} ⟶^{h_{i}} N_{j}^{1}

\underline{Case 2}: a_{j} ∉ N_{j}^{1}.

Then tp(a_{j}, N_{j}^{1}, M_{j}^{1}) is not algebraic, extending the minimal type p_{0} ∈ 𝒮 (N_{0}). Also by clause (viii) of 5.5 there is c ∈ N_{i}\N_{j} which realizes p_{0}. As p_{0} ∈ 𝒮 (N) is minimal

h_{j} (tp(c, N_{j}, N_{i})) = tp(a_{j}, N_{j}^{1}, M_{j}^{1})

so acting as in Case 1 we can also guarantee h_{i}(c) = a_{j}, so a_{j} ∈ Rang(h_{i}) = N_{i}^{1} as required.

∗  ∗  ∗

In the end we have N_{α}^{1} ≤_{𝔎} M_{α}^{1}. If N_{α}^{1} = M_{α}^{1}, then h_{α}^{−1} ↾ M_{0} = h_{α}^{−1} ↾ M_{0}^{1} = h_{α}^{−1} ↾ N_{0}^{1} show that M_{0} can be embedded into N_{α} over N_{0} as required. So assume N_{α}^{1} <_{𝔎} M_{α}^{1}. Now p_{α} ∈ 𝒮 (N_{α}) is inevitable hence h_{α}(p_{α}) ∈ 𝒮 (N_{α}^{1}) is inevitable. Hence some d ∈ M_{α}^{1}\N_{α}^{1} realizes h_{α}(p_{α}) hence d realizes h_{α}(p_{α}) ↾ N_{0}^{1} = p_{0}; also α is a limit ordinal so for some i < α, d ∈ M_{i}^{1} hence for some ζ ∈ S_{i} we have a_{ζ} = d, hence

d = a_{ζ} ∈ N_{ζ+1}^{1} ⊆ N_{α}^{1},

contradicting the choice of d. So we are done. □_{5.6}

5.7 Conclusion. If N ∈ K_{λ} then:

(a) |𝒮 (N)| = λ

(b) there is N_{1}, N <_{𝔎} N_{1} ∈ K_{λ} such that N_{1} is universal over N in K_{λ}

(c) for any regular κ ≤ λ we can demand that (N_{1}, c)_{c∈N} is (λ, κ)-saturated (see 0.5(1))

5.8 Remark. In fact amalgamation in λ and stability in λ (i.e. (a) of 5.7) implies (b) and (c) of 5.7.

5.9 Conclusion. The N ∈ K_{λ^{+}} is saturated above λ (i.e. over models in K_{λ}!).

5.10 Claim. Assume κ = cf(κ) ≤ λ. There are N_{0}, N_{1}, a, N_{0}^{+}, N_{1}^{+} such that

(i) (N_{0}, N_{1}, a) ∈ K_{λ}^{3} and (ii) (N_{0}, N_{1}, a) ≤ (N_{0}^{+}, N_{1}^{+}, a) ∈ K_{λ}^{3} and (iii) (N_{0}^{+}, c)_{c∈N_{0}} is (λ, κ)-saturated, (iv) tp(a, N_{0}, N_{1}) is minimal inevitable and

(v) tp(a, N_{0}^{+}, N_{1}^{+}) is minimal inevitable.

Proof. As in the proof of 5.5 as

E_{2} = {δ : for every i < δ, (N_{δ}^{0}, c)_{c∈N_{i}} is saturated

of cofinality cf(δ)}

is a club of λ^{+}. □_{5.10} 5.11 Claim. 1) In K_{λ} we have disjoint amalgamations.

2) If M ≤_{𝔎} N are in K_{λ} and p ∈ 𝒮 (M) non-algebraic \underline{then} for some N′, c we have: N ≤_{𝔎} N′ ∈ K_{λ} and c ∈ N′\N realizes p.

Proof. 1) First note

⨂ if M ≤_{𝔎} N in K_{λ} we can find α < λ^{+},and sequence ⟨M_{i} : i ≤ α⟩ which is ≤_{𝔎}-increasing continuous, and ⟨a_{i} : i < α⟩ such that (M_{i}, M_{i+1}, a_{i}) is minimal and reduced and N ≤_{𝔎} M_{α}, M = M_{0}. [Why? There is a minimal reduced pair, hence we can find ⟨M_{i} : i < λ^{+}⟩ ≤_{𝔎}-increasing continuous, (M_{i}, M_{i+1}, a_{i}) minimal reduced and M = M_{0}. So by 5.9 we know ⋃_{i<λ^{+}} M_{i} ∈ K_{λ^{+}} is saturated, hence we can embed

N into ⋃_{i<λ^{+}} M_{i} over N so this embedding is into some M_{α}, α < λ^{+}.]

So given M ≤_{𝔎} M^{1}, M^{2}, without loss of generality M^{ℓ} = M_{α_{ℓ}}^{ℓ}, ⟨(M_{i}^{ℓ}, a_{i}^{ℓ}) : i ≤ α_{ℓ}⟩ as above, and start to amalgamate using the extension property and “reduced”.

2) Follows from part (1). □_{5.11}

5.12 Remark. We could prove 5.11 earlier using “reduced triples”. I.e. note that for some ⟨M_{i}^{1} : i < λ^{+}⟩ ∈ Seq_{λ^{+}}[C_{𝔎,λ^{+}}^{1}], for each i for some a the triple (M_{i}^{1}, M_{i+1}^{1}, a) ∈ K_{λ}^{3} is reduced. Hence if M ≤_{𝔎} N from K_{λ}, for some M̄ = ⟨M_{i} : i ≤ α⟩, ≤_{𝔎}- increasing continuous, ⟨M_{i}, M_{i+1}, b_{i}) ∈ K_{λ}^{3} is reduced, M_{0} = M, N ≤ M_{α} ∈ K_{λ} (otherwise find ⟨M_{i}^{2} : i < λ^{+}⟩ ∈ Seq_{λ^{+}}[C_{𝔎,λ^{+}}^{1}] with (M_{i}, M_{i+1}) ≅ (M, N), hence

M^{1} = ⋃_{i<λ^{+}} M_{i}^{1}, M^{2} = ⋃_{i<λ^{+}} M_{i}^{2} are non-isomorphic members of K_{λ^{+}}, contradic-

tion). Now prove by induction on β ≤ α that if M ≤_{𝔎} N_{0} ∈ K_{λ} then N_{0}, M_{β} has disjoint amalgamation over M_{0} = M (i.e. we need to decompose only one side).

\underline{5.13 Question}: If M ∈ K_{λ}, p ∈ 𝒮 (M) is minimal, is it reduced? Or at least, if M_{0} ≤_{𝔎} M_{1} are in K_{λ}, p_{1} ∈ 𝒮 (M_{ℓ}) is non-algebraic, p_{0} = p_{1} ↾ M_{0}, p_{0} is minimal and reduced is also p_{1} reduced? Probably true and would somewhat simplify our work, but not needed for our aims (here and in [Sh 600]). Now 5.5 is an approximation. Can prove it if λ < λ^{ℵ_{0}} or there are E.M. models.

§6 Proving for 𝔎 categorical in λ^{+2}

6.1 Hypothesis. Assume the model theoretic assumptions from 4.1 + 5.1 and so the further model theoretic properties deduced in §4 + §5. We use heavily 4.8.

6.2 Definition. 1) We say (M_{0}, M_{1}, M_{2}) ∈ K_{λ}^{uniq} (has unique (disjoint) amalgamation in K_{λ}) when

(a) M_{0}, M_{1}, M_{2} ∈ K_{λ}

(b) M_{0} ≤_{𝔎} M_{1} and M_{0} ≤_{𝔎} M_{2}

(c) \underline{if} for i = 1, 2 we have g_{ℓ}^{i} : M_{ℓ} → N_{i} ∈ K_{λ} such that:

(i) g_{ℓ}^{i} a ≤_{𝔎}-embedding,

(ii) g_{0}^{i} ⊆ g_{1}^{i} and g_{0}^{i} ⊆ g_{2}^{i} for i = 1, 2

(iii) Rang(g_{1}^{i}) ∩ Rang(g_{2}^{i}) = Rang(g_{0}^{i}) (disjoint amalgamation) for i = 1, 2

\underline{then} we can find N ∈ K_{λ} and ≤_{𝔎}-embeddings

f^{i} : N_{i} → N for i = 1, 2

\underline{such that}

⋀_{ℓ<3} f^{1} ∘ g_{ℓ}^{1} = f^{2} ∘ g_{ℓ}^{2}.

2) Let K_{λ}^{2,uq} is the class of pairs (M_{0}, M_{2}) such that M_{0} ≤_{𝔎} M_{2} are both in K_{λ} and [M_{0} ≤_{𝔎} M_{1} ∈ K_{λ} ⇒ (M_{0}, M_{1}, M_{2}) ∈ K_{λ}^{uniq}] and let K_{λ}^{3,uq} be the class of pairs (M_{0}, M_{2}) ∈ K_{λ}^{2,uq} satisfying M_{0} ≠ M_{2}.

6.3 Claim. 1) If (M_{0}, M_{1}, M_{2}) ∈ K_{λ}^{uniq} \underline{then}

(a) (M_{0}, M_{2}, M_{1}) ∈ K_{λ}^{uniq}

(b) if M_{0} ≤_{𝔎} M′_{2} ≤_{𝔎} M_{2} \underline{then} (M_{0}, M_{1}, M′_{2}) ∈ K_{λ}^{uniq}.

2) Assume M_{0} ≤_{𝔎} M_{2} are from K_{λ} and M_{1} ∈ K_{λ} is universal over M_{0}. \underline{Then} (M_{0}, M_{2}) ∈ K_{λ}^{2,uq} ⇔ (M_{0}, M_{1}, M_{2}) ∈ K_{λ}^{uniq}.

Proof. 1)a) Trivial.

b) Chase arrows (using disjoint amalgamation; i.e. 5.11).

2) Follows by 6.3(1)(a)+(b) and the definition. □_{6.3} 6.4 Lemma. Suppose

⨂ there is (M_{0}, M_{1}, M_{2}) ∈ K_{λ}^{uniq} such that M_{0} ≠ M_{2} and M_{1} is universal

over M_{0}.

\underline{Then}: there are N^{0} <_{𝔎} N^{1} in K_{λ^{+}} such that:

(a) N^{0} ≠ N^{1}

(b) for every c ∈ N^{1}\N^{0} there is M = M_{c} satisfying N^{0} ≤_{𝔎} M ≤_{𝔎} N^{1} and N^{0} ≠ M and c ∈ N^{1}\M.

Proof. Choose ⟨N_{i}^{0} : i < λ^{+}⟩, a sequence of members of K_{λ} which is ≤_{𝔎}-increasing continuous, such that:

(N_{i}^{0}, N_{i+1}^{0}) ≅ (M_{0}, M_{2}).

So N_{i}^{0} ≠ N_{i+1}^{0} hence N_{0} = ⋃_{i<λ^{+}} N_{i}^{0} ∈ K_{λ^{+}} and without loss of generality

|N_{0}| = λ^{+}. We now choose by induction on i < λ^{+}, N_{i}^{1} and M_{i,c} for c ∈ N_{i}^{1}\N_{i}^{0} such that:

(a) N_{i}^{0} ≤_{𝔎} N_{i}^{1} ∈ K_{λ} and N_{i}^{0} ≠ N_{i}^{1}

(b) N_{i}^{1} is ≤_{𝔎}-increasing continuous in i

(c) j < i ⇒ N_{j}^{1} ∩ N_{i}^{0} = N_{j}^{0}; moreover N_{i}^{1} ∩ |N_{0}| = N_{i}^{0}

(d) N_{i}^{0} ≤_{𝔎} M_{i,c} ≤_{𝔎} N_{i}^{1}

(e) c ∉ M_{i,c}

(f) N_{i}^{0} ≠ M_{i,c}

(g) if j < i and c ∈ N_{j}^{1}\N_{j}^{0} \underline{then} M_{i,c} ∩ N_{j}^{1} = M_{j,c}.

\underline{For i = 0}: Choose N_{i}^{1} such that N_{i}^{0} ≤_{𝔎} N_{i}^{1}, (N_{i}^{1}, c)_{c∈N_{i}^{0}} saturated (any cofinality will do) then by disjoint amalgamation easy to define the M_{0,c} (remembering clause (c)).

\underline{For i limit}: Straightforward.

\underline{For i = j + 1}: First we disjointly amalgamate getting N′_{i} ∈ K_{λ} such that N_{i}^{0} ≤ N′_{i}, N_{j}^{1} ≤_{𝔎} N′_{i} and |N′_{i}| ∩ |N_{0}| = |N_{i}^{0}| (as set of elements). Let N_{i}^{1} be such that:

N′_{i} ≤_{𝔎} N_{i}^{1} ∈ K_{λ}
(N_{i}^{1}, c)_{c∈N′_{i}} is saturated (any cofinality will do)
|N_{i}^{1}| ∩ |N_{0}| = |N_{i}^{0}|.

Lastly we shall find the M_{i,c}’s, the point is that (N_{j}^{0}, N_{i}^{0}, N_{j}^{1}) ∈ K_{λ}^{uniq} (by 6.3(2)).

By ⨂ and Claim 6.3 we could have done the amalgamation in two steps and use uniqueness. Then by uniqueness of saturated extensions embed the result inside N_{i}^{1} and similarly deal with new c’s.

Now let N_{1} =: ⋃_{1<λ^{+}} N_{i}^{1} and for c ∈ N_{1}\N_{0} let M_{c} = ⋃{M_{i,c} : c ∈ N′_{i}} they are

as required. □_{6.4}

Remark. The proof of 6.5 below is like [Sh 88, proof of 3.8 stage(c)]. The aim is to contradict that under I(λ^{+}, 𝔎) = 0 there \underline{are} maximal triples.

6.5 Conclusion. Assume 𝔎 has amalgamation in λ^{+}. If ⨂ of 6.4, \underline{then} there is no maximal triple (M, N, a) in K_{λ^{+}}^{3}.

Proof. We can get by 6.4 a contradiction. [Why? Assume (N_{0}, N_{2}, a) ∈ K_{λ^{+}}^{3} maximal, (N^{0}, N^{1}) as in the conclusion (i.e. (a) + (b)) of 6.4; by categoricity in λ^{+} without loss of generality N_{0} = N^{0} and let N_{1} = N^{1}. Now 𝔎 has amalgamation for λ^{+} so there are N ∈ K_{λ^{+}} and f such that f : N_{2} → N is a ≤_{𝔎}-embedding of N_{2} into N over N_{0} and N_{1} ≤_{𝔎} N. If f(a) ∉ N_{1}, \underline{then} (N_{0}, N_{2}, a) <_{f} (N_{1}, N, f(a)) contradict maximality. If f(a) ∈ N_{1}, then M_{f(a)} is well defined (see 6.4) and (N_{0}, N_{2}, a) <_{f} (M_{f(a)}, N, f(a)) contradicts maximality.] □_{6.5}

6.6 Remark. 1) Another proof is to replace the assumption “𝔎 has amalgamation in λ^{+}” by I(λ^{+}, K) < 2^{λ^{+2}}. We start with N_{0}, N_{1}, N_{2}, a as above and building, for every S ⊆ λ^{+2}, a sequence ⟨M_{α}^{S} : α < λ^{+2}⟩ of members of K_{λ^{+}}, which is ≤_{𝔎}-increasing continuous, and α ∈ S ⇒ (M_{α}^{S}, M_{α+1}^{S}, a_{α}^{S}) ≅ (N_{0}, N_{2}, a), and α ∈ λ^{+2}\S ⇒ (M_{α}^{S}, M_{α+1}^{S}) ≅ (N_{0}, N_{1}) which are as in (a) + (b) of 6.4. Let M^{S} = ⋃_{α<λ^{+2}} M_{α}^{S} ∈ K_{λ^{+2}} and from M^{S}/ ≅ we can reconstruct S/𝒟_{λ^{+2}}. So here we use

I(λ^{+2}, K) < 2^{λ^{+2}} but not the definitional weak diamond for λ^{++}.

2) Note that if 2^{λ^{+}} < 2^{λ^{++}} then the assumption of 6.6(1) implies the assumption of 6.5.

6.7 Claim. Assume

(∗) 2^{λ} < 2^{λ^{+}} < 2^{λ^{++}} (or at least the definitional weak diamond for λ^{+}, λ^{++}).

If ⨂ of 6.4 fails, we get I(λ^{+2}, K) ≥ μ_{wd}(λ).

Proof. By 3.23 and 6.8 below.

The following serves to prove 6.7 6.8 Claim. Assume M ∈ K_{λ} ⇒ |𝒮 (M)| ≤ λ.

If K_{λ}^{3,uq} = ∅ (see Definition 6.2(2)), \underline{then} there is F, an amalgamation choice function F for C = C_{𝔎,λ^{+}}^{0} with the weak λ^{+}-coding property.

Proof. The point is that if M̄ = ⟨M_{α} : α < λ^{+}⟩ ∈ Seq_{λ^{+}}[C] and α < λ^{+}, M_{α} <_{𝔎} N ∈ K_{λ}, \underline{then} for some β ∈ (α, λ^{+}) we have:

M_{β} is universal over M_{α}, so as K_{λ}^{3,uq} = ∅ necessarily
(M_{α}, M_{β}, N) ∉ K_{λ}^{uniq} and rest should be clear.

Of course, we use the extension property. □_{6.8}

6.9 Remark. We can work in the context of §3, we need the existence of a saturated (equivalently super limit) M ∈ K_{λ^{+}}. We now say how to replace μ_{wd}(λ^{+2}) by 2^{λ^{+2}}.

6.10 Claim. 1) Assume each M ∈ K_{λ^{+}} is saturated above λ.

If (M, N, a) ∈ K_{λ}^{3}, it and every (M′, N′, a) ∈ K_{λ}^{3} above it has the extension property but for every (M′′, N′′, a) ≥ (N, N, a) (all in K_{λ}^{3}) for some M^{∗} ≥_{𝔎} M′′ from K_{λ}, in amalgamation (M^{∗}, N^{∗}, a) ≥ (M′′, N′′, a) the type of M^{∗} ∪ N inside N^{∗} is not determined \underline{then} some F (actually F^{∗}) has the λ^{+}-coding.

2) If above we just require that the type of M^{∗} ∪ N′′ inside N^{∗} is not determined, \underline{then} some F (actually F^{∗}) has weak λ^{+}-coding.

3) We can restrict ourselves to disjoint embedding.

6.11 Discussion. 1) We get IE(λ^{+2}, 𝔎) = 2^{λ^{+2}} when (2^{λ^{+}})^{+} < 2^{λ^{+2}}. See more in [Sh 600].

We now prove 0.2

6.12 Theorem. Assume (∗) of 6.7 (or at least the conclusion of 6.7). \underline{Then} I(λ^{+2}, K) = 1 ⇒ I(λ^{+3}, K) > 0.

Remark. As in [Sh 88, §3].

Proof. By 0.19(1) it is enough to show that for some M ∈ K_{λ^{++}} there is M′, M ≤_{𝔎} M′ ∈ K_{λ^{++}}, M ≠ M′. [Why? As then we can choose by induction on i < λ^{+3} models M_{i} ∈ K_{λ^{+2}}, ≤_{𝔎}-increasing continuous, M_{i} ≠ M_{i+1}, for i = 0 use K_{λ^{+2}} ≠ ∅, for i limit take union, for i = j + 1 use the previous sentence; so M_{λ^{+3}} = ∪{M_{i} : i < λ^{+3}} ∈ K_{λ^{+3}} as required.] By 6.7, the statement ⨂ of 6.4 holds so we can find (N^{0}, N^{1}) as there so by 6.5 there is in K_{λ^{+}}^{3} no maximal member. This implies (easy, see 2.6(6)) that there are M^{∗} ≤_{𝔎} N^{∗} from K_{λ^{+2}} such that M^{∗} ≠ N^{∗} which as said above (by categoricity in λ^{+}), suffices. □_{6.12}

§7 Extensions and conjugacy

7.1 Hypothesis. Assume the model theoretic assumptions from 4.1 + 5.2 and the further model theoretic properties deduced since then (but not in 6.7,6.12) or just

(a) 𝔎 abstract elementary class

(b) 𝔎 has amalgamation in λ

(c) 𝔎 is categorical in λ (can be weakened)

(d) 𝔎 is stable in λ (see 5.7, clause (a))

(e) there is an inevitable p ∈ 𝒮 (N) for N ∈ K_{λ} (holds by 5.3)

(f) the basic properties in type theory.

We now continue toward eliminating the use of I(λ^{++}, K) = 1 (in 6.12), and give more information. We first deal with the nice types in 𝒮 (N), N ∈ K_{λ} in particular the realize/materialize problem which is here: if N_{1} ≤_{𝔎} N_{2} are in K_{λ}, p_{ℓ} ∈ 𝒮 (N_{ℓ}) is minimal, p_{1} ≤ p_{2}, are they conjugate? (i.e. does p_{2} ∈ 𝒮_{p_{1}}(N_{2})?).

7.2 Claim. If N ∈ K_{λ} and p ∈ 𝒮 (N) is minimal and reduced or just p is reduced (see Definition 2.3(7)), \underline{then} p is inevitable.

Proof. Suppose N, p form a counterexample. We can then find N_{1} and a such that N ≤_{𝔎} N_{1} ∈ K_{λ}, a ∈ N_{1}\N and p = tp(a, N, N_{1}) and (N, N_{1}, a) is reduced. As p is not inevitable, there is N_{2} such that: N ≤_{𝔎} N_{2} ∈ K_{λ}, N ≠ N_{2} but no element of N_{2} realizes p. By amalgamation in K_{λ}, without loss of generality there is N_{3} ∈ K_{λ} such that ℓ ∈ {1, 2} ⇒ N_{ℓ} ≤_{𝔎} N_{3}. By 5.3 (i.e.7.1(e)) there is q ∈ 𝒮 (N) which is inevitable so there are c_{ℓ} ∈ N_{ℓ} with q = tp(c_{ℓ}, N, N_{ℓ}) for ℓ ∈ {1, 2}. By the equality of types (and amalgamation in K_{λ}) there is N^{+} ∈ K, a ≤_{𝔎}-extension of N_{1} and a ≤_{𝔎}-embedding f of N_{2} into N^{+} over N such that f(c_{2}) = c_{1}; so without loss of generality N^{+} = N_{3} and f is the identity, hence c_{1} = c_{2}. Now a ∉ N_{2} as p = tp(a, N, N_{1}) is not realized in N_{2}. So (N, N_{1}, a) ≤ (N_{2}, N_{3}, a) and N_{2} ∩ N_{1}\N ≠ ∅ contradicting “(N, N_{1}, a) is reduced”. □_{7.2}

7.3 Claim. 1) If κ = cf(κ) ≤ λ and N̄ = ⟨N_{i} : i ≤ ωκ⟩ is an ≤_{𝔎}-increasingly continuous sequence, N_{i} ∈ K_{λ}, N_{i+1} universal over N_{i}, and p ∈ 𝒮 (N_{ωκ}) is minimal reduced (or minimal inevitable) \underline{then} for some i < ωκ we have p ↾ N_{i} ∈ 𝒮 (N_{i}) is minimal (so p is the unique, non-algebraic extension of p ↾ N_{i} in 𝒮 (N_{ωκ}) (and of course, there is one)).

2) If λ ≥ κ = cf(κ), N̄ = ⟨N_{i} : i ≤ κ⟩ is ≤_{𝔎}-increasing continuous in K_{λ} and p ∈ 𝒮 (N_{κ}) is minimal and reduced and the set Y =: {i < κ : N_{i+1} is (λ, κ)-saturated over N_{i}} is unbounded in κ \underline{then} for every large enough i ∈ Y there is an isomorphism f from N_{i+1} onto N_{κ} which is the identity on N_{i} and

(∗) f maps p ↾ N_{i+1} ∈ 𝒮 (N_{i+1}) to p ∈ 𝒮 (N_{κ}).

Hence as p is minimal reduced, so is p ↾ N_{i+1}.

Proof. 1) We can choose (N_{i}^{0}, N_{i}^{1}, a) ∈ K_{λ}^{3} for i < λ^{+} reduced, ≤-increasing continuous such that N_{i}^{0} ≠ N_{i+1}^{0}. Let N_{ℓ} = ⋃_{i<λ^{+}} N_{i}^{ℓ}. As in the proof of 5.5 for

c ∈ N_{1}\N_{0}

I_{c}^{∗} = {j < λ^{+} : c ∈ N_{j}^{1} and tp(c, N_{j}^{0}, N_{j}^{1}) is minimal}

is empty or is an end segment of λ^{+} and

E = \{δ < λ^{+} : \underline{if} c ∈ N_{δ}^{1} and I_{c}^{∗} ≠ ∅ \underline{then} I_{c}^{∗} ∩ δ

is an unbounded subset of δ; and if α < δ

then for some β ∈ (α, δ), N_{β}^{0} is universal over N_{α}^{0} and if P r is one of the properties reduced and/or inevitable and/or minimal and there is i ≥ δ such that

(N_{i}^{0}, N_{i}^{1}, c) has P r, then there are arbitrarily

large such i < δ}

is a club of λ^{+}; for the universality demand in the definition of E use categoricity in λ^{+}. Let δ ∈ acc(acc(E)), cf(δ) = κ, let ⟨α_{ζ} : ζ < ωκ⟩ be an increasing continuous sequence of ordinals from E with limit δ, now set α_{ωκ} = δ and N′_{ζ} =: N_{α_{ζ}}^{0}.

So there is an isomorphism f from N_{ωκ} onto N_{α_{ωκ}}^{0} such that for every ζ < ωκ we have N_{α_{2ζ}}^{0} ≤_{𝔎} f(N_{α_{2ζ}}) ≤_{𝔎} N_{α_{2ζ+1}}^{0} (so if ζ is a limit ordinal, then N_{α_{ζ}}^{0} = N_{α_{2ζ}}^{0} = f(N_{ζ})), so without loss of generality f is the identity. As p ∈ 𝒮 (N_{ωκ}) is inevitable (by assumption or by 7.2) and N_{ωκ} = N_{α_{ωκ}}^{0} <_{𝔎} N_{α_{ωκ}}^{1}, for some c ∈ N_{α_{ωκ}}^{1}\N_{α_{ωκ}}^{0} we have p = tp(c, N_{α_{ωκ}}^{0}, N_{α_{ωκ}}^{1}), so for some β < α_{ωκ} we have c ∈ N_{β}^{1}. As p is minimal (by assumption) clearly δ ∈ I_{c}, but δ ∈ E so Min(I_{c}) < δ, but I_{c} is an end segment of λ^{+} hence without loss of generality for some ζ < ωκ we have β = α_{ζ} ∈ I_{c}. So for ξ ∈ (ζ, ωκ), both p ∈ 𝒮 (N_{ωκ}) and p ↾ N_{ξ} ∈ 𝒮 (N_{ξ}) are non-algebraic extensions of the minimal p ↾ N_{α_{ζ}}^{0} ∈ 𝒮 (N_{α_{ζ}}^{0}) and N_{α_{ζ}}^{0} ≤_{𝔎} N_{ξ} ≤_{𝔎} N_{ωκ}, all in K_{λ}, so we have proved part (1).

2) Without loss of generality every ζ < κ is in Y. We can find ⟨N′_{ζ} : ζ ≤ ωκ⟩ as in part (1), moreover satisfying “N′_{ζ+1} is (λ, κ)-saturated over N_{ζ}” and such that: for every ζ ≤ κ we have N_{ζ} = N′_{ωζ}. So again choose ζ < κ as there, we set β = α_{ωζ} ∈ I_{c}^{∗}. If ξ ∈ Y & ωξ > ζ clearly by the uniqueness of (λ, κ)- saturated models there is an isomorphic f from N_{ξ+1} = N′_{ω(ξ+1)} onto N_{κ} = N′_{ωκ} over N_{ξ} = N′_{ωκ}, and f(p ↾ N_{ξ+1}) = p is proved as above. □_{7.3} 7.4 Claim. 1) If M_{0} ≤_{𝔎} M_{1} are in K_{λ} and the types p_{ℓ} ∈ 𝒮 (M_{ℓ}) are minimal reduced, for ℓ = 0, 1 and p_{0} = p_{1} ↾ M_{0} \underline{then} p_{0}, p_{1} are conjugate; (i.e. there is an isomorphic f from M_{0} onto M_{1} such that f(p_{0}) = p_{1}).

2) If in addition M ≤_{𝔎} M_{0} and M_{0}, M_{1} are (λ, κ)-saturated over M, \underline{then} p_{0}, p_{1} are conjugate over M.

Remark. Note that p minimal (or reduced) implies that p is not algebraic.

Proof. 1) Let ⟨(N_{i}^{0}, N_{i}^{1}, a) : i < λ^{+}⟩ and E be as in the proof of 7.3 and κ = cf(κ) ≤ λ. For each δ ∈ S_{κ} =: {α < λ^{+} : α ∈ E and cf(α) = κ}, and minimal reduced p ∈ 𝒮 (N_{δ}^{0}), we know that for some i_{p} < δ, p ↾ N_{i_{p}}^{0} is minimal reduced [why? by 7.3(1),(2)] and some q_{p} ∈ 𝒮 (N_{i_{p}}^{0}) is conjugate to p. For κ = cf(κ) ≤ λ, q ∈ 𝒮 (N_{i}^{0}), i < λ^{+}, r ∈ 𝒮 (N_{i}^{0}) minimal let

A_{q,r}^{κ,i} = {δ < λ^{+} : there is a type p such that r ⊆ p ∈ 𝒮 (N_{δ}^{0}), p non-algebraic (this determines p), p minimal reduced, i_{p} = i, q_{p} = q

(and clearly p ↾ N_{i}^{0} = r) and cf(δ) = κ}.

Next let

E_{1} = {δ < λ^{+} : for every κ = cf(κ) ≤ λ,
r, q ∈ 𝒮 (N_{i}^{0}) and i < δ, \underline{if} A_{q,r}^{κ,i} is well defined and
unbounded in λ^{+} \underline{then} it is unbounded in δ\}.

So if δ_{1} ∈ E_{1}, κ = cf(δ_{1}), p_{1} ∈ 𝒮 (N_{δ_{1}}^{0}) is minimal reduced, \underline{then} we can find δ_{0} < δ_{1}, cf(δ_{0}) = κ, and p_{0} ∈ 𝒮 (N_{δ_{0}}^{0}) minimal reduced with q_{p_{1}} = q_{p_{0}}, i_{p_{1}} = i_{p_{0}}, p_{0} ↾ N_{i_{p_{0}}}^{0} = p_{1} ↾ N_{i_{p_{1}}}^{0} call it r, it is necessarily minimal.

As p_{1}, p_{0} extend r, N_{i_{p_{0}}}^{0} = N_{i_{p_{1}}}^{0} ≤_{𝔎} N_{δ_{0}}^{0} ≤_{𝔎} N_{δ_{1}}^{0}, necessarily p_{1} = p_{0} ↾ N_{δ_{0}}^{0}, and also they are both conjugate to q_{p_{0}} = q_{p_{1}} hence they are conjugate. Next we prove

(∗) if M_{0} <_{𝔎} M_{1} are in K_{λ}, M_{1} is (λ, κ)-saturated over M_{0}, p′_{0} ∈ 𝒮 (M_{0}) is minimal reduced and p′_{0} ≤ p′_{1} ∈ 𝒮 (M_{1}), p′_{1} non-algebraic, \underline{then} p′_{0}, p′_{1} are conjugate.

Above we have a good amount of free choice in choosing p_{1} ∈ 𝒮 (N_{δ_{1}}^{0}) (it should be minimal and reduced) so we could have chosen p_{1} to be conjugate to p′_{0} i.e. is in 𝒮_{p′_{0}}(N_{δ_{1}}^{0}); now also the corresponding p_{0} is conjugate to p_{1} hence p_{0} is conjugate to p′_{0}, hence we can find an isomorphism f_{0} from M_{0} onto N_{δ_{0}}^{0}, f_{0}(p′_{0}) = p_{0}, and extend it to an isomorphism f_{1} from M_{1} onto N_{δ_{1}}^{0}, so necessarily f(p′_{1}) = p_{1} (as p_{1} is the unique non-algebraic extension of p_{0} in 𝒮 (M_{δ_{1}})). As p_{0}, p_{1} are conjugate through f_{1} also p′_{0}, p′_{1} are conjugate. So (∗) holds.

Now assume just M_{0} ≤_{𝔎} M_{1} are in K_{λ}, p_{0} ∈ 𝒮 (M_{0}) minimal reduced, p_{1} ∈ 𝒮 (M_{1}) the unique non-algebraic extension of p_{0} and it is reduced (and necessarily minimal). There is M_{2}, M_{1} ≤_{𝔎} M_{2} ∈ K_{λ}, M_{2} is (λ, κ)-limit over M_{1} hence also over M_{0} and let p_{2} be the unique non-algebraic extension of p_{1} in 𝒮 (M_{2}) hence p_{2} is also the unique non-algebraic extension of p_{0} in 𝒮 (M_{2}). Using (∗) on (M_{0}, M_{2}, p_{0}, p_{2}) and on (M_{1}, M_{2}, p_{1}, p_{2}) and get that p_{0}, p_{2} are conjugate and that p_{1}, p_{2} are conjugate hence p_{1}, p_{2} are conjugate, the required result.

2) Similar proof. □_{7.4}

7.5 Claim. 1) Assume M_{1} ≤_{𝔎} M_{2} are in K_{λ} and M_{2} is (λ, κ)-saturated over M_{1}. If p_{1} ∈ 𝒮 (M_{1}) is minimal and reduced, \underline{then} p_{2}, the unique non-algebraic extension of p_{1} in 𝒮 (M_{2}), is reduced (and, of course, minimal).

2) There is no need to assume “p_{1} reduced”.

Proof. 1) We can find ⟨N_{i} : i ≤ κ⟩, an ≤_{𝔎}-increasingly continuous sequence in K_{λ} such that N_{i+1} is (λ, κ)-saturated over N_{i} and N_{κ} = M_{1}. So by 7.3(1),(2), for some ζ < κ we have: p_{2} ↾ N_{ζ} is minimal and for some isomorphism f from N_{ζ+1} onto N_{κ} we have f(p_{1} ↾ N_{ζ}) = p_{1} and f ↾ N_{ζ} = id_{N_{ζ}}. Also M_{1}, N_{ζ+1} are isomorphic over N_{ζ} (as both are (λ, κ)-saturated over it) hence there is an isomorphism g from N_{ζ+1} onto M_{2} over N_{ζ}. Now p_{1} = f(p_{1} ↾ N_{ζ+1}) and f_{2} =: g(p_{1} ↾ N_{ζ+1}) are nonalgebraic extensions of p_{1} ↾ N_{ζ} which are minimal, hence p_{1} = p_{2} ↾ M_{1} and p_{2} is as mentioned in 7.5. Now g ∘ f^{−1} show that p_{1}, p_{2} are conjugate so as p_{1} is reduced also p_{2} is reduced.

2) Easy as we can find N, M_{1} ≤_{𝔎} N, q ∈ 𝒮 (N) extend p_{1} and is minimal reduced; without loss of generality N ≤_{𝔎} M_{2} and M_{2} is (λ, κ)-saturated over N, and apply part (1). □_{7.5}

7.6 Claim. Assume

(a) N_{i,j} ∈ K_{λ} for i ≤ δ_{1}, j ≤ δ_{2}

(b) ⟨N_{i,j} : j ≤ δ_{2}⟩ is ≤_{𝔎}-increasingly continuous for each i ≤ δ_{1}

(c) ⟨N_{i,j} : i < δ_{1}⟩ is ≤_{𝔎}-increasingly continuous for each j ≤ δ_{2}

(d) ⟨N_{i,j} : i ≤ δ_{1}, j ∈ δ_{2}⟩ is smooth, i.e. N_{i_{1},j_{1}} ∩N_{i_{2},j_{2}} = N_{min{i_{1},i_{2}}}∩N_{min{j_{1},j_{2}}} (e) N_{i+1,j+1} is universal over N_{i,j+1} ∪ N_{i+1,j} (i.e. N_{i+1,j+1} is universal over

some N′_{i+1,j+1} where N_{i,j+1} ∪ N_{i+1,j} ⊆ N′_{i+1,j+1}),

(f) δ_{1} is divisible by cf(δ_{2}) × λ × ω (and even easier if δ_{1} = 1!).

\underline{Then} N_{δ_{1},δ_{2}} is (λ, cf(δ_{1}))-saturated over N_{i,δ_{2}} for i < δ_{1}.

Proof. Without loss of generality δ_{2} = cf(δ_{2}). (Why? Let ⟨α_{ε} : ε ≤ cf(δ_{2})⟩ be increasingly continuous with limit δ_{2} such that [ε limit ↔ α_{ε} limit], and use N′_{i,ε} = N_{i,α_{ε}}). So δ_{1} is divisible by δ_{2} × λ × ω. For i < δ_{1}, j < δ_{2} let M_{i,j}, M′_{i,j} be such that M_{i+1,j} ∪ M_{i,j+1} ⊆ M_{i,j} ≤_{𝔎} M′_{i,j} ≤ M_{i+1,j+1} and M′_{i,j} is (λ, cf(λ))-saturated over M_{i,j}.

Now let p ∈ 𝒮 (N) be minimal and reduced; and for i ≤ δ_{1}, j ≤ δ_{2} let p_{i,j} ∈ 𝒮 (N_{i,j}) be the unique non-algebraic extension of p in 𝒮 (N_{i,j}) so it is minimal. Now for i < δ_{1}, note that ⟨M′_{i+ε,ε} : ε < δ_{2}⟩ is ≤_{𝔎}-increasing (not continuous!) and M′_{i+ε+1,ε+1}

is (λ, cf(λ))-saturated over M′_{i+ε,ε} and ⋃_{ε<δ_{2}} M′_{i+ε,ε} = ⋃_{ε<δ_{2}} N_{i+ε+1,ε+1} = N_{i+δ_{2},δ_{2}},

hence by 7.5 we know that p_{i+δ_{2},δ_{2}} is reduced (and minimal). In fact, similarly α < δ_{1} & cf(α) = cf(δ_{2}) ⇒ p_{α,δ_{2}} is reduced. As N_{i+1,j+1} ≠ N_{i+1,j} ∪ N_{i,j+1} and clause (d) (smoothness) necessarily N_{i+δ_{2},δ_{2}} < N_{i+δ_{2}+1,δ_{2}}, hence some c ∈ N_{i+δ_{2}+1,δ_{2}}\N_{i+δ_{2},δ_{2}} realizes p_{i+δ_{2},δ_{2}}. So if α ≤ δ_{1} is divisible by δ_{2} × λ and has cofinality cf(δ_{2}) and β < α, then by 5.6 N_{α,δ_{2}} is universal over N_{β,δ_{2}}. As δ_{1} is divisible by cf(δ_{2}) × λ × ω we are done. □_{7.6}

7.7 Lemma. 1) For every N ∈ K_{λ^{+}} we can find a representation N̄ = ⟨N_{i} : i < λ^{+}⟩, with N_{i+1} being (λ, cf(λ))-saturated over N_{i}.

2) If for ℓ = 1, 2 we have N^{ℓ} = ⟨N_{i}^{ℓ} : i < λ^{+}⟩ as in part (1) \underline{then} there is an isomorphism f from N^{1} onto N^{2} mapping N_{i}^{1} onto N_{i}^{2} for each i < λ^{+}. Moreover for any i < λ^{+} and isomorphism g from N_{i}^{1} onto N_{i}^{2} we can find an isomorphism f from N^{1} onto N^{2} extending g and mapping N_{j}^{1} onto N_{j}^{2} for each j ∈ [i, λ^{+}).

3) If N^{0} ≤_{𝔎} N^{1} are in K_{λ^{+}} \underline{then} we can find representations N̄^{ℓ} of N^{ℓ} as in (1) with N_{i}^{0} = N^{0} ∩ N_{i}^{1}, (so N_{i}^{0} ≤_{𝔎} N_{i}^{1}).

4) For any strictly increasing function f : λ^{+} → λ^{+}, we can find N_{i,ε} for i < λ^{+}, ε ≤ λ × (1 + f(i)) such that:

(a) N_{i,ε} ∈ K_{λ}

(b) ⟨N_{i,ε} : ε ≤ λ × (1 + f(i))⟩ is strictly ≤_{𝔎}-increasing continuous

(c) for each ε, ⟨N_{i,ε} : i ∈ [i_{ε}, λ^{+})⟩ is a representation as in (1) where i_{ε} = Min{i : ε ≤ λ × (1 + f(i))}

(d) if ε < λ × (1 + f(i)) and i < j < λ^{+} \underline{then} N_{j,ε} ∩ N_{i,λ×(1+f(i))} = N_{i,ε}

(e) N_{i+1,ε+1} is (λ, ℵ_{0})-saturated over N_{i+1,ε} ∪ N_{i,ε+1}.

Proof. Straight.

4) First use f′ : λ^{+} → λ^{+} which is f′(i) = λ^{ω} × f(i). Then define the N_{i,ε} (ε < λ × (1 + f′(i); i < λ^{+}). “Forget” about “N_{i+1,ε} is (λ, cf(λ))-saturated over N_{i,ε}”, remember we have disjoint amalgamation by 5.11. Now by 7.6, even for ε limit divisible by λ^{3} we get N_{i+λ,ε} is (λ, cf(λ))-saturated over N_{i,ε}, so renaming all is O.K.). □_{7.7}

We can deduce from ?, but to keep the door open to other uses we shall not use

7.8 Claim. If κ_{ℓ} = cf(κ_{ℓ}) ≤ λ, and N_{ℓ} is (λ, κ_{ℓ})-saturated over N for ℓ = 1, 2 \underline{then} N_{1}, N_{2} are isomorphic over N.

Proof. We can define by induction on i ≤ λ × κ_{1}, and then by induction on j ≤ λ × κ_{2}, M_{i,j} such that:

(a) M_{i,j} ∈ K_{λ}

(b) M_{0,0} = N

(c) i_{1} ≤ i & j_{1} ≤ j ⇒ M_{i_{1},j_{1}} ≤_{𝔎} M_{i,j}

(d) M_{i_{1},j_{1}} ∩ M_{i_{2},j_{2}} = M_{min{i_{1},i_{2}},min{j_{1},j_{2}}} (e) M_{i,j} is ≤_{𝔎}-increasing continuous in i

(f) M_{i,j} is <_{𝔎}-increasing continuous in j

(g) M_{0,j} ≠ M_{0,j+1}

(h) M_{i+1,0} ≠ M_{i,0}

(i) M_{i+1,j+1} is universal over M_{i+1,j} ∪ M_{i,j+1}.

There is no problem by 5.11(1) (using the existence of disjoint amalgamation). Now M_{λ×κ_{1},λ×κ_{2}} is the union of the strictly <_{𝔎}-increasing sequence ⟨M_{0,0}⟩ˆ ⟨M_{λ×i,λ×κ_{2}} : i < κ_{1}⟩ hence by 7.6 is (λ, κ_{1})-saturated over M_{0,0} = N hence M_{λ×κ_{1},λ×κ_{2}} ≅_{N} N_{1}. Similarly M_{λ×κ_{1},λ×κ_{2}} is the union of the strictly ≤_{𝔎}-increasing sequence ⟨M_{0,0}⟩ˆ ⟨M_{λ×κ_{1},λ×j} : j < κ_{2}⟩ hence is (λ, κ_{2})-saturated over M_{0,0} = N, hence M_{λ×κ_{1},λ×κ_{2}} ≅_{N} N_{2}. Together N_{1}, N_{2} are isomorphic over N.

□_{7.8}

§8 Uniqueness of amalgamation in 𝔎_{λ}

We deal in this section only with K_{λ}. We want to, at least, approximate unique amalgamation using as starting point ⨂ of 6.4 (see also 6.7), i.e. K_{λ}^{3,uq} ≠ ∅.

8.1 Hypothesis. 1) Assume hypothesis 7.1, so

(a) 𝔎 abstract elementary class

(b) 𝔎 has amalgamation in λ

(c) 𝔎 is categorical in λ (can be weakened)

(d) 𝔎 is stable in λ (see 5.7, clause (a))

(e) there is an inevitable p ∈ 𝒮 (N) for N ∈ K_{λ} (holds by 5.3)

(f) the basic properties in type theory.

2) (M^{∗}, N^{∗}) is some pair in K_{λ}^{3,uq} = {(M_{0}, M_{2}) : M_{0} ≤_{𝔎} M_{2} are in K_{λ} and for every M_{1}, M_{0} ≤_{𝔎} M_{1} ∈ K_{λ} ⇒ (M_{0}, M_{1}, M_{2}) ∈ K_{λ}^{uniq}; equivalently for some M_{1}, (M_{0}, M_{1}, M_{2}) are as in ⨂ of 6.4} (eventually the choice does not matter; if each time instead of ≅ (M^{∗}, N^{∗}) we write ∈ K_{λ}^{2,uq}, see 8.11; but if we start with this definition then the uniqueness theorems will be more cumbersome).

8.2 Definition. Assume δ̄ = ⟨δ_{1}, δ_{2}, δ_{3}⟩, δ_{2} a limit ordinal < λ^{+} but δ_{1}, δ_{3} are < λ^{+} and may be 1. We say that NF_{λ,δ̄}(N_{0}, N_{1}, N_{2}, N_{3}) (we say N_{1}, N_{2} are saturated and smoothly amalgamated in N_{3} over N_{0} for δ̄) when:

(a) N_{ℓ} ∈ K_{λ} for ℓ ∈ {0, 1, 2, 3}

(b) N_{0} ≤_{𝔎} N_{ℓ} ≤_{𝔎} N_{3} for ℓ = 1, 2

(c) N_{1} ∩ N_{2} = N_{0} (i.e. in disjoint amalgamation)

(d) N_{1} is (λ,cf(δ_{1}))-saturated over N_{0}

(e) N_{2} is (λ,cf(δ_{2}))-saturated over N_{0}, if δ_{1} = 1 this means just N_{0} ≤_{𝔎} N_{2} (f) there are N_{1,i}, N_{2,i} for i ≤ λ × δ_{1} (called the witness) such that:

(α) N_{1,0} = N_{0}, N_{1,λ×δ_{1}} = N_{1} (β) N_{2,0} = N_{2} (γ) ⟨N_{ℓ,i} : i ≤ λ × δ_{1}⟩ is <_{𝔎}-increasing continuous for ℓ = 1, 2 (δ) (N_{1,i}, N_{1,i+1}) ≅ (M^{∗}, N^{∗}) (ε) N_{2,i} ∩ N_{1} = N_{1,i} (ζ) N_{3} is (λ,cf(δ_{3}))-saturated over N_{2,λ×δ_{1}}; if δ_{3} = 1 this means just

N_{2,λ×δ_{1}} ≤_{𝔎} N_{3}

\underline{Discussion}: Why this definition of NF? We need a nonforking notion with the usual properties. We first describe a version depending on ⟨δ_{0}, δ_{1}, δ_{2}⟩ and get NF = NF_{λ,δ̄}, δ̄ works like a scaffold – eventually δ̄ disappears.

8.3 Definition. 1) We say N_{1} N_{3} ; ⋃ ; N_{0} N_{2} (or N_{1}, N_{2} are smoothly amalgamated over

N_{0} inside N_{3} or NF_{λ}(N_{0}, N_{1}, N_{2}, N_{3})) \underline{if} we can find M_{ℓ} ∈ K_{λ} (for ℓ < 4) such that:

(a) NF_{λ,⟨λ,λ,λ⟩}(M_{0}, M_{1}, M_{2}, M_{3})

(b) N_{ℓ} ≤_{𝔎} M_{ℓ} for ℓ < 4

(c) N_{0} = M_{0}

(d) M_{1}, M_{2} are (λ, cf(λ))-saturated over N_{0} (follows by (a) see clauses (d), (e) of 8.2).

8.4 Claim. 1) Assume δ̄ = ⟨δ_{1}, δ_{2}, δ_{3}⟩, δ_{ℓ} a limit ordinal < λ^{+} and N_{ℓ} ∈ K_{λ} for ℓ < 3, and N_{1} is (λ, cf(δ_{1}))- saturated over N_{0} and N_{2} is (λ,cf(δ_{2}))-saturated over N_{0} and N_{0} ≤_{𝔎} N_{1}, N_{0} ≤_{𝔎} N_{2} and for simplicity N_{1} ∩ N_{2} = N_{0}. \underline{Then} we can find N_{3} such that NF_{λ,δ̄}(N_{0}, N_{1}, N_{2}, N_{3}).

2) Moreover, we can choose any ⟨N_{1,i} : i ≤ λ × δ_{1}⟩ as in 8.2(f)(α), (γ), (δ) as part of the witness.

Proof. Straightforward (remembering ? (and uniqueness of the (λ, cf(δ_{1}))-saturated model over N_{0})). □_{8.4}

8.5 Claim. In Definition 8.2, if δ_{3} is a limit ordinal, \underline{then} without loss of generality (even without changing ⟨N_{1,i} : i ≤ λ × δ_{1}⟩)

(g) N_{2,i+1} is (λ,cf(δ_{2}))-saturated over N_{i+1}^{1} ∪ N_{i}^{2} (which means it is (λ,cf(δ_{2}))-saturated over some N, where N_{i+1}^{1} ∪ N_{i}^{2} ⊆ N ≤_{𝔎} N_{2,i+1}).

Proof. So assume NF_{λ,δ̄}(N_{0}, N_{1}, N_{2}, N_{3}) holds as witnessed by ⟨N_{ℓ,i} : i ≤ λ × δ_{ℓ}⟩ for ℓ = 1, 2. Now we choose by induction on i ≤ λ × δ_{1} a model M_{2,i} ∈ K_{λ} such that:

(i) N_{2,i} ≤ M_{2,i} (ii) M_{2,0} = N_{2} (iii) M_{2,i} is ≤_{𝔎}-increasing continuous (iv) M_{2,i} ∩ N_{2,λ×δ_{1}} = N_{2,i} moreover M_{2,i} ∩ N_{3} = N_{2,i}

(v) M_{2,i+1} is (λ,cf(δ_{2}))-saturated over M_{2,i} ∪ N_{2,i+1}.

There is no problem to carry the definition. Let M_{3} be such that M_{2,λ×δ_{1}} ≤_{𝔎} M_{3} ∈ K_{λ} and M_{3} is (λ,cf(δ_{3}))-saturated over M_{2,λ×δ_{1}}. So both M_{3} and N_{3} are (λ,cf(δ_{3}))-saturated over N_{2,λ×δ_{1}}, hence they are isomorphic over N_{2,λ×δ_{1}} so let f be an isomorphism from M_{3} onto N_{3} which is the identity over N_{2,λ×δ_{1}}. Clearly ⟨N_{1,i} : i ≤ λ × δ_{1}⟩, ⟨f(M_{2,i}) : i ≤ λ × δ_{1}⟩ are also witnesses for NF_{λ,δ̄}(N_{0}, N_{1}, N_{2}, N_{3}) satisfying the extra demand (g). □_{8.5} 8.6 Claim. (Weak Uniqueness).

Assume that for x ∈ {a, b}, we have NF_{λ,δ̄^{x}}(N_{0}^{x}, N_{1}^{x}, N_{2}^{x}, N_{3}^{x}) as witnessed by ⟨N_{1,i}^{x} : i ≤ λ × δ_{1}^{x}⟩, ⟨N_{2,i}^{x} : i ≤ λ × δ_{1}^{x}⟩ and δ_{1} =: δ_{1}^{a} = δ_{1}^{b}, cf(δ_{2}^{a}) = cf(δ_{2}^{b}) and cf(δ_{3}^{a}) = cf(δ_{3}^{b}) ≥ ℵ_{0}.

Suppose further that f_{ℓ} is an isomorphism from N_{ℓ}^{a} onto N_{ℓ}^{b} for ℓ = 0, 1, 2, moreover: f_{0} ⊆ f_{1}, f_{0} ⊆ f_{2} and f_{1}(N_{1,i}^{a}) = N_{1,i}^{b}.

\underline{Then} we can find an isomorphism f from N_{3}^{a} onto N_{3}^{b} extending f_{1} ∪ f_{2}.

Proof. Without loss of generality N_{2,i+1}^{x} is (λ, cf(δ_{2}))-saturated over N_{1,i+1}^{x} ∪ N_{2,i}^{x} (by 8.5, note the “without changing the N_{1,i}’s”). Now we choose by induction on i ≤ λ × δ_{1} an isomorphism g_{i} from N_{2,i}^{a} onto N_{2,i}^{b} such that: g_{i} is increasing in i and g_{i} extends (f_{1} ↾ N_{1,i}^{a}) ∪ f_{2}. For i = 0 choose g_{0} = f_{2} and for i limit let g_{i} be ⋃_{j<i} g_{j} and for i = j + 1 use

(N_{1,i}, N_{1,i+1}) ≅ (M^{∗}, N^{∗}) (see 8.2) and the extra saturation clause (g). Now we can extend g_{λ×δ_{1}} to an isomorphism from N_{3}^{a} onto N_{3}^{b} as N_{3}^{x} is (λ, cf(δ_{3}))-saturated from N_{2,λ×δ_{1}}^{x} (for x ∈ {a, b}); note that knowing 8.6 possibly the choice of ⟨N_{1,i} : i ≤ λ × δ_{1}⟩ matters. □_{8.6}

Now we prove an “inverted” uniqueness 8.7 Claim. Suppose that

(a) for x ∈ {a, b} we have NF_{λ,δ̄^{x}}(N_{0}^{x}, N_{1}^{x}, N_{2}^{x}, N_{3}^{x})

(b) δ̄^{x} = ⟨δ_{1}^{x}, δ_{2}^{x}, δ_{3}^{x}⟩, δ_{1}^{a} = δ_{2}^{b}, δ_{2}^{a} = δ_{1}^{b},cf(δ_{3}^{a}) = cf(δ_{3}^{b}) all limit ordinals

(c) f_{0} is an isomorphism from N_{0}^{a} onto N_{0}^{b}

(d) f_{1} is an isomorphism from N_{1}^{a} onto N_{2}^{b}

(e) f_{2} is an isomorphism from N_{2}^{a} onto N_{1}^{b}

(f) f_{0} ⊆ f_{1} and f_{0} ⊆ f_{2}.

\underline{Then} there is an isomorphism from N_{3}^{a} onto N_{3}^{b} extending f_{1} ∪ f_{2}.

Before proving 8.8 Subclaim. 1) For any limit ordinals δ_{1}^{a}, δ_{2}^{a}, δ_{3}^{a} ≤ λ we can find M_{i,j} (for i ≤ λ × δ_{1}^{a} and j ≤ λ × δ_{2}^{a}) and M_{3} such that:

(A) M_{i,j} ∈ K_{λ} (B) i_{1} ≤ i_{2} & j_{1} ≤ j_{2} ⇒ M_{i_{1},j_{1}} ≤_{𝔎} M_{i_{2},j_{2}} (C) if i ≤ λ × δ_{1} is a limit ordinal and j ≤ λ × δ_{2} \underline{then} M_{i,j} = ⋃_{ζ<i} M_{ζ,j}

(D) if i ≤ λ × δ_{1} and j ≤ λ × δ_{2} is a limit ordinal \underline{then} M_{i,j} = ⋃_{ξ<j} M_{i,ξ}

(E) (M_{0,j}, M_{0,j+1}) ≅ (M^{∗}, N^{∗})

(F) (M_{i,0}, M_{i+1,0}) ≅ (M^{∗}, N^{∗})

(G) for i_{1}, i_{2} ≤ λ × δ_{1} and j_{1}, j_{2} ≤ λ × δ_{2} we have M_{i_{1},j_{1}} ∩ M_{i_{2},j_{2}} = M_{min{i_{1},i_{2}},min{j_{1},j_{2}}},

(H) M_{λ×δ_{1},λ×δ_{2}} ≤_{𝔎} M_{3} ∈ K_{λ} moreover M_{3} is (λ, cf(δ_{3}))-saturated over M_{λ×δ_{1},λ×δ_{2}}.

2) Moreover, it is O.K. if ⟨M_{0,j} : j ≤ λ × δ_{2}^{a}⟩, ⟨M_{i,0} : i ≤ λ × δ_{1}^{a}⟩ are pregiven as long as both ≤_{𝔎}-increasing continuous in 𝔎_{λ} satisfying (E) + (F) and M_{0,λ×δ_{2}^{a}} ∩ M_{λ×δ_{2}^{a},0} = M_{0,0}.

Proof. 1) This is done by induction on i and for fixed i by induction on j. For i = 0, j = 0 let M_{0,0} ∈ K_{λ} be arbitrary, for i = 0, j limit use clause (D), for i = 0, j = ξ + 1 use clause (E). For i limit use clause (C) (and check). For i = ζ + 1, if j = 0 use clause (F), if j limit use clause (D) and if j = ξ + 1 use the existence of disjoint amalgamation (i.e. 5.11).

Lastly, choose M_{3} ∈ K_{λ} which is (λ, cf(δ_{3}^{a}))-saturated over M_{λ×δ_{1}^{a},λ×δ_{2}^{a}}.

2) Similarly. □_{8.8}

Proof of 8.7. Let M_{i,j}, M_{3} be as in 8.8. For x ∈ {a, b} as NF_{λ,δ̄^{x}}(N_{0}^{x}, N_{1}^{x}, N_{2}^{x}, N_{3}^{x}), we know that there are witnesses ⟨N_{1,i}^{x} : i ≤ λ × δ_{1}^{x}⟩, ⟨N_{2,i}^{x} : i ≤ λ × δ_{1}^{x}⟩ for this, so ⟨N_{1,i}^{x} : i ≤ λ × δ_{1}^{x}⟩ is ≤_{𝔎}-increasing continuous and (N_{1,i}^{x}, N_{1,i+1}^{x}) ≅ (M^{∗}, N^{∗}). So ⟨N_{1,i}^{a} : i ≤ λ × δ_{1}^{a}⟩ is ≤_{𝔎}-increasing continuous sequences with each successive pair isomorphic to (M^{∗}, N^{∗}) hence by 8.8(2) without loss of generality there is an isomorphism g_{1} from N_{1,λ×δ_{1}^{a}}^{a} onto M_{λ×δ_{1}^{a}}, mapping N_{1,i}^{a} onto M_{i,0}; remember N_{1,λ×δ_{1}^{a}}^{a} = N_{1}^{a}. Let g_{0} = g_{1} ↾ N_{0}^{a} = g_{1} ↾ N_{1,0}^{a} so g_{0} ∘ f_{0}^{−1} is an isomorphism from N_{0}^{b} onto M_{0,0}.

As δ_{1}^{b} = δ_{2}^{a}, using 8.8(2) fully without loss of generality there is an isomorphism g_{2} from N_{1,λ×δ_{2}^{a}}^{b} onto M_{0,λ×δ_{2}^{a}} mapping N_{1,j}^{b} onto M_{0,j} (for j ≤ λ × δ_{2}^{a}) and g_{2} extends g_{0} ∘ f_{0}^{−1}. Now we want to use the weak uniqueness 8.6 and for this note:

(α) NF_{λ,δ̄^{a}}(N_{0}^{a}, N_{1}^{a}, N_{2}^{a}, N_{3}^{a}) as witnessed by ⟨N_{1,i}^{a} : i ≤ λ × δ_{1}^{a}⟩, ⟨N_{2,i}^{a} : i ≤ λ × δ_{1}⟩ [why? an assumption]

(β) NF_{λ,δ̄^{a}}(M_{0,0}, M_{λ×δ_{1}^{a},0}, M_{0,λ×δ_{2}^{a}}, M_{3}) as witnessed by the sequences ⟨M_{i,0} : i ≤ λ × δ_{1}⟩, ⟨M_{i,λ×δ_{2}^{a}} : i ≤ λ × δ_{2}^{a}⟩ [why? check]

(γ) g_{0} is an isomorphism from N_{0}^{a} onto M_{0,0} [why? see its choice]

(δ) g_{1} is an isomorphism from N_{1}^{a} onto M_{λ×δ_{1}^{a},0} mapping N_{1,i}^{a} onto M_{i,0} for i ≤ λ × δ_{1}^{a} and extending g_{0} [why? see the choice of g_{1} and of g_{0}]

(ε) g_{2} ∘ f_{2} is an isomorphism from N_{2}^{a} onto M_{0,λ×δ_{2}^{a}} extending g_{0} [why? f_{2} is an isomorphism from N_{2}^{a} onto N_{1}^{b} and g_{2} is an isomorphism

from N_{1}^{b} onto M_{0,λ×δ_{1}^{a}} extending g_{0} ∘ f_{0}^{−1} and f_{0} ⊆ f_{2}].

So there is by 8.6 an isomorphism g_{3}^{a} from N_{3}^{a} onto M_{3} extending g_{1} and g_{2} ∘ f_{2}.

We next want to apply 8.6 to the N_{1}^{b}’s; so note:

(α)′ NF_{λ,δ̄^{b}}(N_{0}^{b}, N_{1}^{b}, N_{2}^{b}, N_{3}^{b}) as witnessed by ⟨N_{1,i}^{b} : i ≤ λ × δ_{2}^{a}⟩, ⟨N_{2,i}^{b} : i ≤ λ × δ_{2}^{a}⟩

(β)′ NF_{λ,δ̄^{b}}(M_{0,0}, M_{0,λ×δ_{2}^{a}}, M_{λ×δ_{1}^{a},0}, M_{3}) as witnessed by the sequences ⟨M_{0,j} : j ≤ λ × δ_{2}^{a}⟩, ⟨M_{λ×δ_{1}^{a},j} : j ≤ λ × δ_{1}^{a}⟩

(γ)′ g_{0} ∘ (f_{0})^{−1} is an isomorphism from N_{0}^{b} onto M_{0,0}

[why? Check.] (δ)′ g_{2} is an isomorphism from N_{1}^{b} onto M_{0,λ×δ_{2}^{a}} mapping N_{1,j}^{b} onto M_{0,j} for

j ≤ λ × δ_{2}^{a} and extending g_{0} ∘ (f_{1})^{−1}

[why? see the choice of g_{2}: it maps N_{1,j}^{b} onto M_{0,j}]

(ε)′ g_{1} ∘ (f_{1})^{−1} is an isomorphism from N_{2}^{b} onto M_{λ×δ_{0}^{a}} extending g_{0} [why? remember f_{1} is an isomorphism from N_{1}^{a} onto N_{2}^{b} extending f_{0} and

the choice of g_{1}: it maps N_{1}^{a} onto M_{λ×δ_{1,0}^{a}}].

So there is an isomorphism g_{3}^{b} form N_{3}^{b} onto M_{3} extending g_{2}, f_{1} ∘ (f_{1})^{−1}. Lastly (g_{3}^{b})^{−1} ∘ g_{3}^{a} is an isomorphism from N_{3}^{a} onto N_{3}^{b} (chase arrows). Also

((g_{b}^{b})^{−1} ∘ g_{3}^{a}) ↾ N_{1}^{a} = (g_{3}^{b})^{−1}(g_{3}^{a} ↾ N_{1}^{a})
= (g_{3}^{b})^{−1}g_{1} = ((g_{3}^{b})^{−1} ↾ M_{λ×δ_{1,0}^{a}}) ∘ g_{1}
= (g_{3}^{b} ↾ N_{2}^{b})^{−1} ∘ g_{1} = ((g_{1} ∘ (f_{1})^{−1})^{−1}) ∘ g_{1}
= (f_{1} ∘ (g_{1})^{−1}) ∘ g_{1} = f_{1}.

Similarly ((g_{3}^{b})^{−1} ∘ g_{3}^{a}) ↾ N_{2}^{a} = f_{2}. So we have finished. □_{8.7}

8.9 Claim. [Uniqueness]. Assume for x ∈ {a, b} we have NF_{λ,δ̄^{x}}(N_{0}^{x}, N_{1}^{x}, N_{2}^{x}, N_{3}^{x}) and cf(δ_{1}^{a}) = cf(δ_{1}^{b}), cf(δ_{2}^{a}) = cf(δ_{2}^{b}), cf(δ_{3}^{a}) = cf(δ_{3}^{b}), all δ_{ℓ}^{x} limit ordinals.

If f_{ℓ} is an isomorphism from N_{ℓ}^{a} onto N_{ℓ}^{b} for ℓ < 3 and f_{0} ⊆ f_{1}, f_{0} ⊆ f_{2} \underline{then} there is an isomorphism f from N_{3}^{a} onto N_{3}^{b} extending f_{1}, f_{2}.

Proof. Let δ̄^{c} = ⟨δ_{1}^{c}, δ_{2}^{c}, δ_{3}^{c}⟩ = ⟨δ_{2}^{a}, δ_{1}^{a}, δ_{3}^{a}⟩; by 8.4 there are N_{ℓ}^{c} (for ℓ ≤ 3) such that NF_{λ,δ̄^{c}}(N_{0}^{c}, N_{1}^{c}, N_{2}^{c}, N_{3}^{c}). There is for x ∈ {a, b} an isomorphism g_{0}^{x} from N_{0}^{a} onto N_{0}^{c} (as K_{λ} is categorical in λ) and without loss of generality g_{0}^{b} = g_{0}^{a} ∘ f_{0}. Similarly for x ∈ {a, b} there is an isomorphism g_{1}^{x} from N_{1}^{x} onto N_{2}^{c} extending g_{0}^{x} (as N_{1}^{x} is (λ, cf(δ_{1}^{x}))-saturated over N_{0}^{x} and also N_{2}^{c} is (λ, cf(δ_{2}^{c}))-saturated over N_{0}^{c} and cf(δ_{2}^{c}) = cf(δ_{1}^{a}) = cf(δ_{1}^{x})) and without loss of genrality g_{1}^{b} = g_{1}^{a} ∘ f_{1}). Similarly for x ∈ {a, b} there is an isomorphism g_{2}^{x} from N_{2}^{x} onto N_{1}^{c} extending g_{0}^{x} (as N_{2}^{x} is (λ, cf(δ_{2}^{x}))-saturated over N_{0}^{x} and also N_{1}^{c} is (λ, cf(δ_{1}^{c}))-saturated over N_{0}^{c} and cf(δ_{1}^{c}) = cf(δ_{2}^{a}) = cf(δ_{2}^{x})) and without loss of generality g_{2}^{b} = g_{2}^{b} ∘ f_{2}. So by 8.7 for x ∈ {a, b} there is an isomorphism g_{3}^{x} from N_{3}^{x} onto N_{3}^{c} extending g_{1}^{x} and g_{2}^{x}. Now (g_{3}^{b})^{−1} ∘ g_{3}^{a} is an isomorphism from N_{3}^{a} onto N_{3}^{b} extending f_{1}, f_{2} as required. □_{8.9}

8.10 Conclusion. [Symmetry].

If NF_{λ,⟨δ_{1},δ_{2},δ_{3}⟩}(N_{0}, N_{1}, N_{2}, N_{3}) \underline{then} NF_{λ,⟨δ_{2},δ_{1},δ_{3}⟩}(N_{0}, N_{2}, N_{1}, N_{3}).

Proof. By 8.7 (and 8.9).

8.11 Claim. 1) In Definition 8.2 we can replace (N_{1,i}, N_{1,i+1}) ≅ (M^{∗}, N^{∗}) by (N_{1,i}, N_{1,i+1}) ∈ K^{3,uq}.

Proof. Like the proof of 8.7 (get (M_{i,0}, M_{i+1,0}) ≅ (M^{∗}, N^{∗}), (M_{0,j}, M_{0,j+1}) ∈ K_{λ}^{3,uq}), but as we shall not use it, we do not elaborate. □_{8.11}

Now we turn to smooth amalgamation (not necessarily saturated, see Definition 8.3).

8.12 Claim. 1) If NF_{λ,δ̄}(N_{0}, N_{1}, N_{2}, N_{3}) each δ_{ℓ} limit \underline{then} NF_{λ}(N_{0}, N_{1}, N_{2}, N_{3}) (see Definition 8.3).

2) In Definition 8.3 we can add:

(d)^{+} M_{ℓ} is (λ, cf(λ))-saturated over N_{0} and moreover over N_{ℓ},

(e) M_{3} is (λ, cf(λ))-saturated over M_{1} ∪ M_{2} (actually this is given by (f)(ζ) of Definition 8.2).

Proof. 1) By 8.8 we can find M_{i,j} for i ≤ λ × (δ_{1} + λ), j ≤ λ × (δ_{2} + λ) for δ̄′ =: ⟨δ_{1} + λ, δ_{2} + λ, δ_{3}⟩ and choose M′_{3} ∈ K_{λ} which is (λ, cf(δ_{3}))-saturated over M_{λ×δ_{1},λ×δ_{2}}. So NF_{λ,δ̄}(M_{0,0}, M_{λ×δ_{1},0}, M_{0,λ×δ_{2}}, M_{3}), hence by 8.9 without loss of generality M_{0,0} = N_{0}, M_{λ×δ_{1},0} = N_{1}, M_{0,λ×δ_{2}} = N_{2}, and N_{3} = M′_{3}. Lastly, let M_{3} be (λ, cf(λ))-saturated over M′_{3}. Now clearly also NF_{λ,⟨δ_{1}+λ,δ_{2}+λ,δ_{3}+λ⟩}(M_{0,0}, M_{λ×(δ_{1}+λ),0}, M_{0,λ×(δ_{2}+λ)}, M_{3}) and N_{0} = M_{0,0}, N_{1} = M_{λ×δ_{2},0} ≤_{𝔎} M_{λ×(δ_{2}+λ),0}, N_{2} = M_{0,λ×δ_{2}} ≤_{𝔎} M_{0,λ×(δ_{2}+λ)} and M_{λ×(δ_{1}+λ),0} is (λ, cf(λ))-saturated over M_{λ×δ_{1},0} and M_{0,λ×(δ_{2}+λ)} is (λ, cf(λ))-saturated over M_{0,λ×δ_{2}} and N_{3} = M′_{3} ≤_{𝔎} M_{3}. So we get all the requirements for NF_{λ}(N_{0}, N_{1}, N_{2}, N_{3}) (as witnessed by ⟨M_{0,0}, M_{λ×(δ_{1}+λ),0}, M_{0,λ×(δ_{2}+λ)}, M_{3}⟩).

2) Similar proof. □_{8.12}

8.13 Claim. [Uniqueness of smooth amalgamation]: If NF_{λ}(N_{0}^{x}, N_{1}^{x}, N_{2}^{x}, N_{3}^{x}) for x ∈ {a, b}, f_{ℓ} an isomorphism from N_{ℓ}^{a} onto N_{ℓ}^{b} for ℓ < 3 and f_{0} ⊆ f_{1}, f_{0} ⊆ f_{2} \underline{then} f_{1} ∪ f_{2} can be extended to a ≤_{𝔎}-embedding of N_{3}^{a} into some ≤_{𝔎}-extension of N_{3}^{b} (so if N_{3}^{x} is (λ, κ)-saturated over N_{1}^{x} ∪ N_{2}^{x} for x = a, b, we can extend f_{1} ∪ f_{2} to an isomorphism from N_{3}^{a} onto N_{3}^{b}.

Proof. For x ∈ {a, b} let the sequence ⟨M_{ℓ}^{x} : ℓ < 4⟩ be a witness to NF_{λ}(N_{0}^{x}, N_{1}^{x}, N_{2}^{x}, N_{3}^{x}) as in 8.3, 8.12(2) so in particular NF_{λ,⟨λ,λ,λ⟩}(M_{0}^{x}, M_{1}^{x}, M_{2}^{x}, M_{3}^{x}). By chasing arrows and uniqueness, i.e. 8.7, without loss of generality M_{ℓ}^{a} = M_{ℓ}^{b} for ℓ < 4 and f_{0} = id_{N_{0}^{a}}. As M_{1}^{a} is (λ, cf(λ))-saturated over N_{1}^{a} and also over N_{1}^{b} and f_{1} is an isomorphism from N_{1}^{a} onto N_{1}^{b}, clearly there is an automorphism g_{1} of M_{1}^{a} such that f_{1} ⊆ g_{1}, hence also id_{N_{0}^{a}} = f_{0} ⊆ f_{1} ⊆ g_{1}. Similarly there is an automorphism g_{2} of M_{1}^{a} extending f_{1} hence f_{0}. So g_{ℓ} ∈ AUT(M_{ℓ}^{a}) for ℓ = 1, 2 and g_{1} ↾ M_{0}^{a} = f_{0} = g_{2} ↾ M_{0}^{a}. By the uniqueness of NF_{λ,⟨λ,λ,λ⟩} (i.e. Claim 8.9) there is an automorphism g_{4} of M_{4}^{a} extending g_{1} ∪ g_{2}. This proves the desired conclusion. □_{8.13}

8.14 Claim. Assume

(a) δ̄ = ⟨δ_{1}, δ_{2}, δ_{3}⟩, δ_{ℓ} < λ^{+} is a limit ordinal for ℓ = 1, 2, 3; N_{0} ≤_{𝔎} N_{ℓ} ≤_{𝔎} N_{3} for ℓ = 1, 2 and

(b) N_{ℓ} is (λ, cf(δ_{ℓ}))-saturated over N_{0}

(c) N_{3} is cf(δ_{3})-saturated over N_{1} ∪ N_{2}.

\underline{Then} NF_{λ}(N_{0}, N_{1}, N_{2}, N_{3}) iff NF_{λ,δ̄}(N_{0}, N_{1}, N_{2}, N_{3}).

Proof. The “if” direction holds by 8.13. For the “only if” direction, by the proof of 8.12(1) (and Definition 8.2, 8.3) we can find M_{ℓ}(ℓ ≤ 3) such that NF_{λ,δ̄}(M_{0}, M_{1}, M_{2}, M_{3}) and clauses (b), (c), (d) of Definition 8.3 hold so by 8.12 also NF_{λ}(M_{0}, M_{1}, M_{2}, M_{3}). Easily there are for ℓ < 3, an isomorphism f_{ℓ} from M_{ℓ} onto N_{ℓ} such that f_{0} = f_{ℓ} ↾ M_{ℓ}. By the uniqueness for smooth amalgamation (i.e. 8.13) we can find an isomorphism f_{3} from M_{3} onto N_{3} extending f_{1} ∪ f_{2}. So as NF_{λ,δ̄}(M_{0}, M_{1}, M_{2}, M_{3}), also NF_{λ,δ̄}, (f_{0}(M_{0}), f_{3}(M_{1}), f_{3}(M_{2}), f_{3}(M_{3})); i.e. NF_{λ,δ̄}(N_{0}, N_{1}, N_{2}, N_{3}) as required. □_{8.14}

8.15 Claim. [Monotonicity]: If NF_{λ}(N_{0}, N_{1}, N_{2}, N_{3}) and N_{0} ≤_{𝔎} N′_{1} ≤_{𝔎} N_{1} and N_{0} ≤_{𝔎} N′_{2} ≤_{𝔎} N_{2} and N′_{1} ∪ N′_{2} ⊆ N′_{3} ≤_{𝔎} N_{3} \underline{then} NF_{λ}(N_{0}, N′_{1}, N′_{2}, N′_{3}).

Proof. Read Definition 8.3.

8.16 Claim. [Symmetry]: NF_{λ}(N_{0}, N_{1}, N_{2}, N_{3}) holds \underline{if and only if} NF_{λ}(N_{0}, N_{2}, N_{1}, N_{3}) holds.

Proof. By Claim 8.10 (and Definition 8.3).

8.17 Claim. Assume α < λ^{+} is an ordinal and for x ∈ {a, b, c} the sequence ⟨N_{i}^{x} : i ≤ α⟩ is a ≤_{𝔎}-increasing sequence of members of K_{λ}, for x = a, b the sequence is ≤_{𝔎}-increasing continuous for i ≤ α, N_{i}^{b}∩N_{α}^{a} = N_{i}^{a}, N_{i}^{c}∩N_{α}^{a} = N_{i}^{a}, N_{i}^{a} ≤_{𝔎} N_{i}^{b} ≤_{𝔎} N_{i}^{c} and N_{i}^{c} is (λ, κ_{i})-saturated over N_{i}^{b} and NF_{λ,δ̄^{i}}(N_{i}^{a}, N_{i+1}^{a}, N_{i}^{c}, N_{i+1}^{b}) (so i < α ⇒ N_{i}^{c} ≤_{𝔎} N_{i+1}^{b}) where δ̄^{i} = ⟨δ_{1}^{i}, δ_{2}^{i}, δ_{3}^{i}⟩ sequence of limit ordinals, i < α ⇒ δ_{2}^{i+1} = δ_{i}^{3}, and for i < 0 limit,

cf(δ_{i}^{3}) = ∑_{j<i} δ_{j}^{3}, δ_{1} = ∑_{β<α} δ_{β}^{1} and δ_{3} = κ_{α}, δ̄ = ⟨δ_{1}, δ_{2}^{0}, δ_{3}⟩. \underline{Then} NF_{λ,δ̄}(N_{0}^{a}, N_{α}^{a}, N_{0}^{b}, N_{α}^{c}).

Proof. Use uniqueness of 8.9, lastly use 8.9 to show N_{α}^{b} is (λ, cf(α))-saturated over N_{α}^{a} ∪ N_{0}^{b}. □_{8.17}

8.18 Claim. Assume that α < λ^{+} and for x ∈ {a, b} we have ⟨N_{i}^{x} : i ≤ α⟩ is ≤_{𝔎}-increasing continuously sequence of members of K_{λ}.

1) If NF_{λ}(N_{i}^{a}, N_{i+1}^{a}, N_{i}^{b}, N_{i+1}^{b}) for each i < α \underline{then} NF_{λ}(N_{0}^{a}, N_{α}^{a}, N_{0}^{b}, N_{α}^{b}).

2) If α_{1} < λ^{+}, α_{2} < λ^{+} and M_{i,j} (i ≤ α_{1}, j ≤ α_{2}) are as in 8.8, and for each i < α_{1}, j < α_{2} we have:

M_{i,j+1} M_{i+1,j+1} ; ⋃ ; M_{i,j} M_{i+1,j}

\underline{then} M_{i,0} M_{α_{1},α_{2}} ; ⋃ ; M_{0,0} M_{0,j} for i ≤ α_{1}, j ≤ α_{2}.

Proof. 1) We first prove special cases and use them to prove more general cases.

\underline{Case A}: N_{i+1}^{a} is (λ, δ_{i}^{1})-saturated over N_{i}^{a} and N_{i+1}^{b} is (λ, δ_{i}^{2})-saturated over N_{i+1}^{a} ∪ N_{i}^{b} for i < α.

We can choose by induction on i ≤ α, N_{i}^{c} ∈ K_{λ} such that

(a) N_{0}^{b} ≤_{𝔎} N_{0}^{c} ≤_{𝔎} N_{i+1}^{b}, N_{0}^{c} is (λ, δ_{0}^{2})-saturated over N_{0}^{b}, and NF_{λ,⟨δ_{1}^{0},δ_{2}^{0},δ_{2}^{0}⟩}(N_{0}^{a}, N_{1}^{a}, N_{0}^{c}, N_{1}^{b})

(b) N_{α}^{c} ∈ K_{λ} is (λ, δ_{3}^{α})-saturated over N_{α}^{b}.

(Possible by uniqueness; i.e. 8.13 and monotonicity, i.e. 8.15). Now we can use 8.17.

\underline{Case B}: For each i < α we have: N_{i+1}^{a} is (λ, κ_{i})-saturated over N_{i}^{a}. Let δ̄^{i} = (κ_{i}, λ, λ). We can find a ≤_{𝔎}-increasing sequence ⟨M_{i}^{x} : i ≤ α⟩ for x ∈ {a, b, c}, continuous for x = a, b such that i < α ⇒ M_{i}^{b} ≤_{𝔎} M_{i}^{c} ≤_{𝔎} M_{i+1}^{b} and M_{α}^{b} ≤_{𝔎} M_{α}^{c} and NF_{λ,δ̄^{i}}(M_{i}^{a}, M_{i+1}^{a}, M_{i}^{c}, M_{i+1}^{b}) by choosing M_{i}^{a}, M_{i}^{b}, M_{i}^{c} by induction on i. By Case A we know that NF_{λ}(M_{0}^{a}, M_{α}^{a}, M_{0}^{b}, M_{α}^{c}) holds.

We can now choose an isomorphism f_{0}^{a} from N_{0}^{a} onto M_{0}^{a} (exists as K is categorical in λ) and then a ≤_{𝔎}-embedding of N_{0}^{b} into M_{0}^{b} extending f_{0}^{a}. Next we choose by induction on i ≤ α, f_{i}^{a} an isomorphism from N_{i}^{a} onto M_{i}^{a} such that: j < i ⇒ f_{j}^{a} ⊆ f_{i}^{a}, possible by “uniqueness of the (λ, κ_{i})-saturated model over M_{i}^{a}”. (0.28)

Now we choose by induction on i ≤ α, a ≤_{𝔎}-embedding f_{i}^{b} of N_{i}^{b} into M_{i}^{b} extending f_{i}^{a} and f_{j}^{b} for j < i. For i = 0 we have done it, for i limit use ⋃_{j<i} f_{j}^{b},

lastly for i a successor ordinal let i = j + 1, now we have

(∗)_{2} NF_{λ}(M_{i}^{a}, M_{i+1}^{a}, f_{i}^{b}(N_{i}^{b}), M_{i+1}^{b})

[why? because NF_{λ,δ̄^{i}}(M_{i}^{a}, M_{i+1}^{a}, M_{i}^{c}, M_{i+1}^{b}) by the choice of the

M_{ζ}^{x}’s hence by 8.14 we have NF_{λ}(M_{i}^{a}, M_{i+1}^{a}, M_{i}^{c}, M_{i+1}^{b}) and as
M_{i}^{a} ≤_{𝔎} f_{i}^{b}(N_{i}^{b}) ≤ M_{i}^{b}, M_{i}^{c} by 8.15 we get (∗)_{2}].

By (∗)_{2} and the uniqueness of smooth amalgamation 8.13 there is f_{i}^{b} as required. So without loss of generality f_{α}^{b} is the identity, so we have N_{0}^{a} = M_{0}^{a}, N_{α}^{a} = M_{α}^{a}, N_{0}^{b} ≤_{𝔎} M_{0}^{b}, N_{α}^{b} ≤_{𝔎} M_{α}^{b}; also as said above NF_{λ}(M_{0}^{a}, M_{α}^{a}, M_{0}^{b}, M_{α}^{b}) holds so by monotonicity i.e. 8.15 we get NF_{λ}(N_{0}^{a}, N_{α}^{a}, N_{0}^{b}, N_{α}^{b}) as required.

\underline{Case C}: General case.

We can find M_{i}^{ℓ} for ℓ < 3, i ≤ α such that:

(a) M_{i}^{ℓ} ∈ K_{λ}

(b) for each ℓ < 3, M_{i}^{ℓ} is ≤_{𝔎}-increasing in i

(c) M_{i}^{0} = N_{i}^{a}

(d) M_{i+1}^{ℓ+1} is (λ, λ)-saturated over M_{i+1}^{ℓ} ∪ M_{i}^{ℓ+1} for ℓ < 2, i < α

(e) NF_{λ}(M_{i}^{ℓ}, M_{i+1}^{ℓ}, M_{i}^{ℓ+1}, M_{i}^{ℓ+1}) for ℓ < 2, i < α

(f) M_{0}^{ℓ+1} is (λ, λ)-saturated over M_{0}^{ℓ}

(g) for ℓ < 2 and i < α limit we have

M_{i}^{ℓ+1} is (λ, λ)-saturated over ⋃_{j<i} M_{j}^{ℓ+1} ∪ M_{i}^{ℓ}

(h) for i < α limit we have

NF_{λ} (⋃_{j<i} M′_{j}, M_{i}^{1}, ⋃_{j<i} M_{j}^{2}, M_{i}^{2}) .

[how? as in the proof of 8.8].

Now note:

(∗)_{4} M_{i}^{ℓ+1} is (λ, cf(λ × (1 + i)))-saturated over M_{i}^{ℓ} [why? If i = 0 by clause (f), if i a successor ordinal by clause (d) and if i is a limit ordinal then by clause (g)]

(∗)_{5} for i < α, NF_{λ}(M_{i}^{0}, M_{i+1}^{0}, M_{i}^{2}, M_{i+1}^{2})

[why? we use Case B for α = 2 with M_{i}^{0}, M_{i+1}^{0}, M_{i}^{1}, M_{i+1}^{1},

M_{i}^{2}, M_{i+1}^{2} here standing for N_{0}^{a}, N_{0}^{b}, N_{1}^{a}, N_{1}^{b}, N_{2}^{a}, N_{2}^{b} there].

Now we continue as in Case B (using f_{i}^{a} = id_{N_{i}^{a}} and defining by induction on i a ≤_{𝔎}-embedding f_{i}^{b} of N_{i}^{b} into M_{i}^{c}).

2) By part (1) for each i the sequences ⟨M_{β,i} : β ≤ α_{1}⟩, ⟨M_{β,i+1} : β ≤ α_{1}⟩ we get

M_{α_{1}} M_{α_{1},i+1} ; ⋃ ; M_{0,i} M_{0,i+1} hence by symmetry (i.e. 8.13) we have M_{0,i+1} M_{α_{1},i+1} ; ⋃ ; M_{0,i} M_{α_{1},i}.

Applying part (1) to the sequences ⟨M_{0,j} : j ≤ α_{2}⟩, ⟨M_{α_{1},j} : j ≤ α_{2}⟩ we get

M_{0,α_{2}} M_{α_{1},α_{2}} ; ⋃ ; M_{0,0} M_{α_{1},0} hence by symmetry (i.e. 8.13) we have M_{α_{1},0} M_{α_{1},α_{2}} ; ⋃ ; M_{0,0} M_{0,α_{2}}; by

monotonicity, i.e 8.15 (or restriction of the matrix) we get the desired conclusion. □_{8.18}

8.19 Conclusion. Assume ⟨N_{i}^{ℓ} : i ≤ α⟩ is ≤_{𝔎}-increasing continuous for ℓ = 0, 1 where N_{i}^{ℓ} ∈ K_{λ} and N_{i+1}^{1} is (λ, κ_{ℓ})-saturated over N_{i+1}^{0} ∪ N_{i}^{1} and NF_{λ}(N_{i}^{0}, N_{i}^{1}, N_{i+1}^{0}, N_{i+1}^{1}).

\underline{Then} N_{α}^{1} is (λ, cf(∑_{i<α} κ_{i}))-saturated over N_{α}^{0} ∪ N_{0}^{1} (if α is a limit ordinal, “N_{i+1}^{1}

is universal over N_{i+1}^{0} ∪ N_{i}^{1}” suffice).

Proof. The case α not limit is trivial so assume α is a limit ordinal. We choose by induction on i ≤ α, a sequence ⟨M′_{i,ε} : ε ≤ ε(i)⟩ such that:

(a) ⟨M_{i,ε} : ε ≤ ε(i)⟩ is (strictly) <_{𝔎}-increasing continuous

(b) N_{i}^{0} ≤_{𝔎} M_{i,ε} ≤_{𝔎} N_{i}^{1}

(c) N_{i}^{0} = M_{i,0}

(d) ε(i) is (strictly) increasing continuous in i,

(e) j < i & ε ≤ ε(j) ⇒ M_{i,ε} ∩ N_{j}^{1} = M_{j,ε}

(f) ε(0) = 1, M_{i,1} = N_{0}^{1}

(g) for i > 0, λ divides ε(i)

(h) N_{ℓ}^{i} ≤_{𝔎} M_{i+1,ε(i)+1}.

If we succeed, then ε(α) is divisible by λ and ⟨M_{i,ε} : ε ≤ ε(α)⟩ is (strictly) <_{𝔎}- increasing continuous, M_{α,0} = N_{α}^{0}, and M_{α,ε(α)} ≤_{𝔎} N_{α}^{1} but it includes N_{i}^{1} for i < α hence (as α is a limit ordinal) it includes ⋃_{i<α} N_{i}^{1} = N_{α}^{1}; and by ? we conclude that

N_{α}^{1} = M_{α,ε(α)} is (λ, cf(α))-saturated over M_{α,1} hence over N_{α}^{0} ∪ N_{0}^{1} (both ≺ M_{α,i}). For i = 0, and i limit there is not much to do. For i successor we use 8.20 below.

8.20 Conclusion. 1) If NF_{λ}(N_{0}, N_{1}, N_{2}, N_{3}) and ⟨M_{0,ε} : ε ≤ ε(∗)⟩ is ≤_{𝔎}-increasing continuous, N_{0} ≤_{𝔎} M_{0,ε} ≤_{𝔎} M_{2} \underline{then} we can find ⟨M_{1,ε} : ε ≤ ε(∗)⟩ and N′_{3} such that:

(a) N_{3} ≤_{𝔎} N′_{3} ∈ K_{λ}

(b) ⟨M_{1,ε} : ε ≤ ε(∗)⟩ is ≤_{𝔎}-increasing continuous

(c) M_{1,ε} ∩ N_{2} = M_{0,ε}

(d) N_{1} ≤_{𝔎} M_{1,ε} ≤_{𝔎} N′_{3}

(e) if M_{0,0} = N_{0} then M_{1,0} = N.

90 SAHARON SHELAH

2) If N_{3} is universal over N_{1} ∪ N_{2}, \underline{then} without loss of generality N_{3} = N_{2}. Proof. 1) Straightforward by uniqueness.

2) Follows by (1). □_{8.19}, □_{8.20}

§9 Nice extensions in K_{λ^{+}}

9.1 Hypothesis. Assume the hypothesis 8.1.

So by §8 we have reasonable control on \underline{smooth} amalgamation in K_{λ}. We use this to define “nice” extensions in K_{λ^{+}}. This is treated again in §10.

9.2 Definition. 1) Let M_{0} ≤_{λ^{+}}^{∗} M_{1} mean:

(a) M_{ℓ} ∈ K_{λ^{+}}, for ℓ = 0, 1

(b) we can find M̄^{ℓ} = ⟨M_{i}^{ℓ} : i < λ^{+}⟩, a representation of M^{ℓ}, so M_{i}^{ℓ} ∈ K_{λ} (and M_{i}^{ℓ} is ≤_{𝔎}-increasing continuously and M_{ℓ} = ⋃_{i<λ^{+}} M_{i}^{ℓ}) such that:

NF_{λ}(M_{i}^{0}, M_{i+1}^{0}, M_{i}^{1}, M_{i+1}^{1}) for i < λ^{+}.

2) Let M_{0} <_{λ^{+},κ}^{+} M_{1} mean M_{0} ≤_{λ^{+}}^{∗} M_{1} by some witnesses M_{i}^{ℓ} (for i < λ^{+}, ℓ < 2) such that NF_{λ,⟨κ,1,κ⟩}(M_{i}^{0}, M_{i+1}^{0}, M_{i}^{1}, M_{i+1}^{1}). If κ = λ, we omit it.

9.3 Claim. 1) If M_{0} ≤_{λ^{+}}^{∗} M_{1} and M̄^{ℓ} = ⟨M_{i}^{ℓ} : i < λ^{+}⟩ is a representation of M_{ℓ} (as in 8.18) \underline{then} for some club E of λ^{+}, for every α < β from E we have NF_{λ}(M_{α}^{0}, M_{β}^{0}, M_{α}^{1}, M_{β}^{1}).

2) Similarly for <_{λ^{+},κ}^{+}; if M_{0} <_{λ^{+},κ_{ℓ}}^{∗} M_{1}, M̄^{ℓ} = ⟨M̄_{i}^{ℓ} : i < λ^{+}⟩ a representation of M_{ℓ} for ℓ = 1, 2 \underline{then} for some club E of λ^{+} for every α < β from E we have NF_{λ,⟨cf(α),1,cf(α)⟩}(M_{α}^{0}, M_{β}^{0}, M_{α}^{1}, M_{β}^{0}).

3) The κ in Definition 9.2(2) does not matter. In fact, if ⟨M_{i}^{ℓ} : i < λ^{+}⟩ are as in 9.2(1) then for some club E of λ^{+} we have: α ∈ E ⇒ M_{α}^{1} ∩ M_{0} = M_{α}^{0} and α < β & α ∈ E & β ∈ E ⇒ [M_{β}^{1} is cf(β)-saturated over M_{β}^{0} ∪ M_{α}^{1}].

Proof. 1) Straightforward by 8.18.

2) Easy.

3) By 8.19. (Could have used 7.8). □_{9.3}

9.4 Claim. 1) For every κ = cf(κ) ≤ λ and M_{0} ∈ K_{λ^{+}} for some M_{1} ∈ K_{λ^{+}} we have M_{0} <_{λ^{+},κ}^{+} M_{1}.

2) <_{λ^{+}}^{∗} are <_{λ^{+},κ}^{+} are transitive.

3) If M_{0} ≤_{𝔎} M_{1} ≤_{𝔎} M_{2} and M_{0} ≤_{λ^{+}}^{∗} M_{2} \underline{then} M_{0} ≤_{λ^{+}}^{∗} M_{1}.

4) [transitivity] If M_{0} ≤_{λ^{+}}^{∗} M_{1} <_{λ^{+},κ}^{+} M_{2} \underline{then} M_{0} <_{λ^{+},κ}^{∗} M_{2}.

Proof. 1) Let ⟨M_{i}^{0} : i < λ^{+}⟩ be a representation of M_{0} such that M_{i+1}^{0} is (λ, κ)-saturated over M_{i}^{0}. We choose by induction on i, M_{i}^{1} ∈ K_{λ} such that ⟨M_{i}^{1} : i < λ^{+}⟩ is <_{𝔎}-increasing continuously, M_{i}^{0} ≤_{𝔎} M_{i}^{1}, M_{i}^{1} ∩ M_{0} = M_{i}^{0} and NF_{λ,⟨κ,1,κ⟩}(M_{i}^{0}, M_{i+1}^{0}, M_{i}^{1}, M_{i+1}^{1}). We can do it by 7.7(4).

2) Concerning <_{λ^{+},κ}^{+} use 9.3 and 8.18 (i.e. transitivity for smooth amalgamations). Now the proof for <_{λ^{+}}^{∗} is similar.

3) By monotonicity for smooth amalgamations; i.e. 8.15.

4) Routine verification. □_{9.4} 9.5 Claim. 1) If M_{0} ≤_{λ^{+}}^{∗} M_{ℓ} for ℓ = 1, 2, κ = cf(κ) ≤ λ and a ∈ M_{2}\M_{0} \underline{then} for some M_{3} and f we have: M_{1} <_{λ^{+},κ}^{+} M_{3} and f is an ≤_{𝔎}-embedding of M_{2} into M_{3} over M_{0} with f(a) ∉ M_{1}, moreover, f(M_{2}) ≤_{λ^{+}}^{∗} M_{3}.

2) [uniqueness] Assume M_{0} <_{λ^{+},κ}^{+} M_{ℓ} for ℓ = 1, 2 \underline{then} there is an isomorphism f from M_{1} onto M_{2} over M_{0}.

Proof. We first prove part (2).

2) By 9.3(1) + (2) there are representations M̄^{ℓ} = ⟨M_{i}^{ℓ} : i < λ^{+}⟩ of M_{ℓ} for ℓ < 3 such that: M_{i}^{ℓ} ∩ M_{0} = M_{0}^{ℓ} and NF_{λ,⟨κ,cf(κ×i),κ⟩}(M_{i}^{0}, M_{i+1}^{0}, M_{i}^{ℓ}, M_{i+1}^{ℓ}).

Now we choose by induction on i < λ an isomorphism f_{i} from M_{i}^{1} onto M_{i}^{2}, increasing with i and being the identity over M_{i}^{0}. For i = 0 use “M_{0}^{ℓ} is (λ, κ)- saturated over M_{0}^{0} for ℓ = 1, 2” which holds by 8.2. For i limit take unions, for i successor ordinal use uniqueness Claim 8.9.

Proof of part (1). Let κ = ℵ_{0}, by 9.4(1) there are for ℓ = 1, 2 models N_{ℓ}^{∗} ∈ K_{λ^{+}} such that M_{ℓ} <_{λ^{+},κ}^{+} N_{ℓ}^{∗}. Now let M̄^{ℓ} = ⟨M_{i}^{ℓ} : i < λ^{+}⟩ be a representation of M_{ℓ} for ℓ = 0, 1, 2 and let N̄^{ℓ} = ⟨N_{i}^{ℓ} : i < λ^{+}⟩ be a representation of N_{ℓ}^{∗} for ℓ = 1, 2. By 9.4(4) and 9.3(3) without loss of generality NF_{λ}(M_{i}^{0}, M_{i+1}^{0}, M_{i}^{ℓ}, M_{i+1}^{ℓ}) for ℓ = 1, 2 and NF_{λ,⟨κ,1,κ⟩}(M_{i}^{ℓ}, M_{i+1}^{ℓ}, N_{i}^{ℓ}, N_{i+1}^{ℓ}) respectively. Now clearly N_{0}^{ℓ} is (λ, κ)-saturated over M_{0}^{ℓ} hence over M_{0}^{0} (for ℓ = 1, 2) so there is an isomorphism f_{0} from N_{0}^{2} onto N_{0}^{1} extending id_{M_{0}^{0}} and f(a) ∉ M_{0}^{1}.

We continue as in the proof of part (2). In the end f = ⋃_{i<λ^{+}} f_{i} is an isomorphism

of N_{2} onto N_{1} over M_{0} and as f_{0}^{1}(a) is well defined and in N_{0}^{1}\M_{0}^{1} clearly f(a) = f_{0}(a) ∉ M_{1}, as required. □_{9.5}

9.6 Claim. If δ is a limit ordinal < λ^{+2} and ⟨M_{i} : i < δ⟩ is a ≤_{λ^{+}}^{∗}-increasing continuous \underline{then} M_{i} ≤_{λ^{+}}^{∗} ⋃_{j<δ} M_{j} for each i < δ.

Proof. We prove it by induction on δ. Now if C is a club of δ with i ∈ C, then we can replace ⟨M_{j} : j < δ⟩ by ⟨M_{j} : j ∈ C⟩ so without loss of generality δ = cf(δ), so δ ≤ λ^{+}; clearly it is enough to prove M_{0} ≤_{λ^{+}}^{∗} ⋃_{j<δ} M_{j}. Let ⟨M_{ζ}^{i} : ζ < λ^{+}⟩ be a

representation of M_{i}.

\underline{Case A}: δ < λ^{+}. Without loss of generality (see 9.3(1)) for every i < j < δ and ζ < λ^{+} we have:

M_{ζ}^{j} ∩ M_{i} = M_{ζ}^{i} and NF_{λ}(M_{ζ}^{i}, M_{ζ+1}^{i}, M_{ζ}^{j}, M_{ζ+1}^{j}). Let M_{ζ}^{δ} = ⋃_{i<δ} M_{ζ}^{i}, so

⟨M_{ζ}^{δ} : ζ < λ^{+}⟩ is ≤_{𝔎}-increasingly continuous sequence of members of K_{λ} with limit M_{δ}, and for i < δ, M_{ζ}^{δ} ∩ M_{i} = M_{ζ}^{i}. By symmetry (see 8.16) we have NF_{λ}(M_{ζ}^{i}, M_{ζ}^{i+1}, M_{ζ+1}^{i}, M_{ζ+1}^{i+1}) so as ⟨M_{ζ}^{i} : i ≤ δ⟩,⟨M_{ζ+1}^{i} : i ≤ δ⟩ are ≤_{𝔎}-increasingly continuous by 8.18 we know NF_{λ}(M_{ζ}^{0}, M_{ζ}^{δ}, M_{ζ+1}^{0}, M_{ζ+1}^{δ}) hence by symmetry (8.16) we have NF_{λ}(M_{ζ}^{0}, M_{ζ+1}^{0}, M_{ζ}^{δ}, M_{ζ+1}^{δ}). So ⟨M_{ζ}^{0} : ζ < λ^{+}⟩, ⟨M_{ζ}^{δ} : ζ < λ^{+}⟩ are witnesses to M_{0} ≤_{λ^{+}}^{∗} M_{δ}.

\underline{Case B}: δ = λ^{+}.

By 9.3(1) (using normality of the club filter, restricting to a club of λ^{+} and renaming), without loss of generality for i < j ≤ 1 + ζ < 1 + ξ < λ^{+} we have

M_{ζ}^{j} ∩ M_{i} = M_{ζ}^{i}, and NF_{λ}(M_{ζ}^{i}, M_{ξ}^{i}, M_{ζ}^{j}, M_{ξ}^{j}). Let us define M_{i}^{λ^{+}} = ⋃_{j<1+i} M_{i}^{j}. So

⟨M_{i}^{λ^{+}} : i < λ^{+}⟩ is a representation of M_{λ^{+}}^{λ} = M_{δ} and continue as before. □_{9.6}

9.7 Claim. Assume M_{0} <_{λ^{+},κ}^{+} M_{2} and a ∈ M_{2}\M_{0}, and for some N ≤_{𝔎} M_{0} we have: N ∈ K_{λ} and tp(a, N, M_{2}) is minimal. \underline{Then} we can find M_{1}, M̄^{0} = ⟨M_{0,i} : i < λ^{+}⟩, M̄^{1} = ⟨M_{1,i} : i < λ^{+}⟩ such that:

(a) M̄^{0} is a ≤_{𝔎}-representation of M_{0}

(b) M̄^{1} is a representation of M_{1}(∈ K_{λ^{+}}), a ∈ M_{1,i}, for all i

(c) M_{0} ≤_{𝔎} M_{1} ≤_{𝔎} M_{2}

(d) for i < λ^{+} we have NF_{λ,⟨λ,1,1⟩}(M_{0,i}, M_{0,i+1}, M_{1,i}, M_{1,i+1}) (hence M_{ℓ,i} = M_{ℓ+1,i} ∩ M_{ℓ})

(e) (M_{0,i}, M_{1,i}, a) ∈ K_{λ}^{3} is minimal and reduced.

Proof. Let ⟨M_{0,i} : i < λ^{+}⟩, ⟨M_{2,i} : i < λ^{+}⟩ be representations of M_{0}, M_{2} respectively, as required in 9.2(2) and without loss of generality N ≤_{𝔎} M_{0,0} and a ∈ M_{2,0}. We now choose by induction on ζ < λ^{+}, an ordinal i(ζ) and models M_{1,i(ζ)}, M_{3,i(ζ)} such that:

(A) i(ζ) < λ^{+} is increasing continuous in ζ and

a ∈ M_{2,i(0)}\M_{0,i(0)}, N ≤_{𝔎} M_{0,i(0)} (B) M_{0,i(ζ)} ≤_{𝔎} M_{1,i(ζ)} ≤_{𝔎} M_{3,i(ζ)} and M_{2,i(ζ)} ≤_{𝔎} M_{3,i(ζ)} (C) a ∈ M_{1,i(0)} and (M_{0,i(ζ)}, M_{1,i(ζ)}, a) is minimal and reduced (D) for ξ < ζ and (ℓ, m) ∈ {(0, 1), (0, 2), (1, 3), (2, 3)} we have

NF_{λ}(M_{ℓ,i(ξ)}, M_{ℓ,i(ζ)}, M_{m,i(ξ)}, M_{m,i(ζ)}) (E) M_{1,i(ζ)}, M_{3,i(ζ)} is ≤_{𝔎}-increasing continuous in ζ.

For ζ = 0 note that for i(0) < λ^{+}, a ∈ M_{2,i(0)} and M_{2,i(0)} is universal over M_{0,i(0)}.

For ζ limit let i(ζ) = ⋃_{ξ<ζ} i(ξ) and M_{1,i(ζ)} = ⋃_{ξ<ζ} M_{1,i(ζ)}.

For ζ = ξ + 1, there is i(ζ) ∈ (i(ξ), λ^{+}) and a model N_{ζ} such that M_{1,i(ξ)} ≤_{𝔎} N_{ζ} ∈ K_{λ} and ≤_{𝔎}-embedding f of M_{0,i(ζ)} into N_{ζ}, f ↾ M_{0,i(ζ)} the identity and (f(M_{0,i(ξ)}), N_{ζ}, a) is minimal and reduced. By uniqueness (i.e. Claim 8.2) we can find such N satisfying (∃M)(N ≤_{𝔎} M ∈ K_{λ} & M_{1,i(ζ)} ≤_{𝔎} M). So we can carry the induction.

Lastly, by uniqueness of <_{λ,κ}^{+} we can make M_{3} = ⋃_{ζ<λ^{+}} M_{3,i(ζ)} to be ≤_{𝔎} M_{2} as

required. □_{9.7} 9.8 Definition. If (M_{0}, M_{1}, a) are as in 9.7 (a)-(e) we say (M_{0}, M_{1}, a) is λ^{+}-locally reduced nice and minimal (λ^{+}-l.r.n.m.). We omit “nice” if we omit clause (d).

9.9 Claim. If (M_{0}, M_{1}, a) is λ^{+}-l.r.n.m. \underline{then} (M_{0}, M_{1}, a) ∈ K_{λ^{+}}^{3} is reduced.

Proof. Check.

Using also ?:

9.10 Claim. M_{0} <_{λ^{+},κ}^{+} M_{1} \underline{if and only if} we can find ⟨M_{j}^{∗}, a_{j} : j < λ^{+} × κ⟩ such that:

(a) M_{j}^{∗} is ≤_{𝔎}-increasing continuous (in K_{λ^{+}})

(b) (M_{j}^{∗}, M_{j+1}^{∗}, a_{j}) is λ^{+}-l.r.n.m.

(c) M_{0}^{∗} = M_{0}, M_{j}^{∗} ∪ M_{j}^{∗} = M_{1}

(d) for some N ≤ M_{0}, N ∈ K_{λ} and minimal reduced p ∈ S(N), for every j, a_{j} realizes p.

Proof. We can find ⟨M_{j}^{∗} : j ≤ λ^{+} × κ) satisfying clauses (a), (b) and (d). By 5.6 easily M_{0}^{∗} <_{λ}^{+} M_{λ^{+}}^{∗}. Now by the uniqueness (= 9.5(2)) + categoricity of K in λ^{+}, we are done. □_{9.10}

9.11 Claim. In (K_{λ^{+}}, <_{λ^{+}}^{∗}) we have disjoint amalgamation.

Proof. First redo 9.5 assuming (M_{0}, M_{ℓ}, a_{ℓ}) for ℓ = 1, 2 is λ^{+}-l.r.n.m., and getting a_{1} ∉ f(M_{2}), f(a_{2}) ∉ M_{1} (and we can start with this). By 9.9 we get M_{1} ∩ f(M_{2}) = M_{3}, so we have disjoint amalgamation. By 9.10 and chasing arrows we get it in general. □_{9.11}

Remark. This is like the proof of disjoint amalgamations in 5.11.

§10 Non-structure for ≤_{λ^{+}}^{∗}

10.1 Hypothesis. Assume the hypothesis 8.1 and the further model theoretic properties deduced since then (including 6.7 but not 6.12.)

It would have been nice to prove all disjoint amalgamations in K_{λ} are NF_{λ}, but this is, at this point, not clear. But as we look upward (i.e. want to prove statement on K_{>λ^{+}}) and <_{λ^{+}}^{∗} is very nice, it will be essentially just as well if for M, N ∈ K_{λ^{+}} we have M ≤_{𝔎} N ⇒ M ≤_{λ^{+}}^{∗} N. Our intention is to assume M^{∗} ≤_{𝔎} N^{∗} is a counterexample of this statement and we would like to say that in a sense this implies the existence of many types over M^{∗} so that we can construct many models in λ^{+2}. Note: building models in K_{λ^{+}}, K_{λ^{++}} by approximations in K_{λ} is nice if we use the smooth amalgamation but we do not have it for non-smooth ones. So we shall use M^{∗} ∈ K_{λ^{+}} being saturated so it has many automorphisms.

10.2 Claim. 1) Assume M_{1} ≤_{𝔎} M_{2} are in K_{λ^{+}}. \underline{Then} we can find M_{0} ∈ K_{λ^{+}} such that M_{0} <_{λ,κ}^{+} M_{1} and M_{0} ≤_{λ^{+}}^{∗} M_{2}.

2) Also we can find ⟨M_{0,i} : i < λ^{+}⟩, an ≤_{𝔎}-increasingly continuous sequence of

members of K_{λ^{+}} such that M_{0,i} <_{λ^{+}}^{∗} M_{0,i+1} and ⋃_{i<λ^{+}} M_{0,i} = M_{1} and

i < λ^{+} ⇒ M_{0,i} ≤_{λ^{+}}^{∗} M_{2}.

Proof. Let (M^{∗}, N^{∗}) ∈ K_{λ}^{3,uq} be from 8.1(2). Let ⟨M_{ℓ,i} : i < λ^{+}⟩ be a representation of M_{ℓ} for ℓ = 1, 2 and without loss of generality M_{ℓ,i+1} is (λ, λ)-saturated over M_{ℓ,i} for ℓ = 1, 2 and M_{2,i} ∩ M_{1} = M_{1,i}.

1) Now choose by induction on i, M_{0,i} such that:

(a) M_{0,i} ≤_{𝔎} M_{1,i}

(b) M_{0,i} is ≤_{𝔎}-increasing continuous

(c) M_{0,i+1} ∩ M_{1,i} = M_{0,i}

(d) M_{1,i+1} is (λ, cf(λ))-saturated over M_{1,i} ∪ M_{0,i+1}

(e) (M_{0,i}, M_{0,i+1}) ≅ (M^{∗}, N^{∗}).

There is no problem to carry the definition. Now let M_{0} = ⋃_{i<λ^{+}} M_{0,i} so M_{0} <_{λ^{+}}^{+} M_{1}

and M_{0} ≤_{λ^{+}}^{∗} M_{2} are checked by their definitions.

2) We choose by induction on i < λ^{+}, ⟨M_{ε,i}^{∗} : ε ≤ 1 + i⟩ such that:

(a) M_{1+i,i}^{∗} = M_{1,λ×(1+ε)×i}

(b) For each ε the sequence ⟨M_{ε,j}^{∗} : ε ≤ j ≤ i⟩ is ≤_{𝔎}-increasing continuous

(c) For each i, the sequence ⟨M_{ε,i}^{∗} : ε ≤ 1 + i⟩ is ≤_{𝔎}-increasing continuous

(d) M_{ε,i}^{∗} ∩ M_{ζ,j}^{∗} = M_{min{ε,ζ},min{i,j}}^{∗}

(e) M_{ε+1,i+1}^{∗} is (λ, cf(λ × (1 + ε)))-saturated over M_{ε,i+1}^{∗} ∪ M_{ε+1,i}^{∗}

(f) NF_{λ,⟨λ,1,λ⟩}(M_{ε,i}^{∗}, M_{ε+1,i}^{∗}, M_{ε,i+1}^{∗}, M_{ε+1,i+1}^{∗})

(g) for ε < 1 + i we have NF_{λ}(M_{ε,i}^{∗}, M_{ε,i+1}^{∗}, M_{2,λ×λ×i}, M_{2,λ×λ×(i+1)}).

For i = 0, i limit no problem, for i = j + 1 first choose N_{i,ε,ζ} ≤_{𝔎} M_{1,λ×λ×i+λ×ζ} for ζ ≤ λ × (1 + i), ≤_{𝔎}-increasing continuous, N_{i,ε,ζ} ∈ K_{λ}, N_{i,ε,0} = M_{1,j}(= M_{1+i,i}^{∗}), (N_{i,ε,ζ}, N_{i,ε,ζ+1}) ≅ (M^{∗}, N^{∗}) and N_{i,ε,ζ+1} ∩ M_{2,λ×λ×i+λ×ζ} = N_{i,ε,ζ}.

Now by 5.6, without loss of generality N_{i,ε,λ} is (λ, λ)-saturated over M_{1,j}, and we choose it as M_{1+j,i}^{∗}, and we choose M_{1,i+1}^{∗} as M_{1+i,i} note that clauses (a) and (f) holds. Now we can find N_{ε,j_{1}} for j_{1} < 1 + j as in 8.8 and use uniqueness of the (λ, λ × (1 + i))-saturated model over M_{1,j}. □_{10.3}

10.3 Conclusion. Assume M ≤_{𝔎} N are from K_{λ^{+}}. If ⟨M_{i} : i < λ^{+}⟩ is ≤_{λ^{+}}^{∗}-increasing continuous and for each i for some N_{i} we have M_{i} <_{λ^{+}}^{+} N_{i} ≤_{λ^{+}}^{∗} M_{i+1} \underline{then} for some (M′, N′) we have:

(M′, N′) ≅ (M, N)

M′ = ⋃_{i<λ} M_{i}

i < λ^{+} ⇒ M_{i} ≤_{λ^{+}}^{∗} N′.

Proof. By 10.2(2) and by the implication M^{a} ≤_{λ^{+}}^{∗} M^{b} ≤_{λ^{+}}^{+} M^{c} ⇒ M^{a} <_{λ^{+}}^{+} M^{c} and by the uniqueness of M′′ over M′ when M′ <_{λ^{+}}^{+} M′′. □_{10.3}

10.4 Lemma. If ≤_{𝔎}↾ K_{λ^{+}} is not ≤_{λ^{+}}^{∗} \underline{then} I(λ^{+2}, K) = 2^{λ^{+2}}.

Proof. Let S ⊆ {δ < λ^{+2} : cf(δ) = λ^{+}} be stationary. We shall below construct a model M_{S} ∈ K_{λ^{+2}} such that from the isomorphism type of M_{1}^{S} we can reconstruct S/𝒟_{λ^{+2}}; this clearly suffices. Choose M ≤_{𝔎} N from K_{λ^{+}} such that ¬(M ≤_{λ^{+}}^{∗} N) so by 9.4(3) without loss of generality |N\M| = λ^{+}.

We choose by induction on α < λ^{+2} a model M_{α}^{S} such that:

(a) M_{α}^{S} ∈ K_{λ^{+}} has universe λ × (1 + α)

(b) for β < α we have M_{β}^{S} ≤_{𝔎} M_{α}^{S}

(c) if α = β + 1, β ∉ S then M_{β}^{S} <_{λ^{+}}^{+} M_{α}^{S}

(d) if α = β + 1, β ∈ S then (M_{β}^{S}, M_{α}^{S}) ≅ (M, N)

(e) if β < α, β ∉ S then M_{β}^{S} ≤_{λ^{+}}^{∗} M_{α}^{S}.

We use freely the transitivity (9.4(4)) and continuity (9.6) of ≤_{λ}^{∗} and [M^{a} ≤_{𝔎} M^{b} ≤_{𝔎} M^{c} in K_{λ}, ¬(M^{a} ≤_{λ^{+}}^{∗} M^{b} ⇒ ¬(M^{a} ≤_{λ^{+}}^{∗} M^{c})] 9.4(3). The cases that α = 0, α is a limit ordinal and α = β + 1, β ∉ S present no problem.

For α = β + 1, β ∈ S so cf(β) = λ^{+}, let ⟨γ_{i} : i < λ^{+}⟩ be increasing continuous with limit β and cf(γ_{i}) ≤ λ^{+}, hence γ_{i} ∉ S. By 9.3(2) without loss of generality M_{γ_{i}} <_{λ^{+}}^{∗} M_{γ_{i+1}}. Now use 10.3 (and the uniqueness (9.5(2))). □_{10.4} 10.5 Conclusion. Assume I(λ^{+2}, K) < 2^{λ^{+2}} (in addition to hypothesis 10.1). \underline{Then} 1) ≤_{λ^{+}}^{∗}=≤_{𝔎}↾ K_{λ^{+}}.

2) (K_{λ^{+}}, ≤_{𝔎}), has disjoint amalgamation so no M ∈ K_{λ^{+2}} is ≤_{𝔎}-maximal.

3) K_{λ^{+3}} ≠ ∅.

Proof. 1) By 10.4.

2) By 9.11 (and part (1)).

3) By 10.5(2) and 2.6(8) with λ there replaced by λ^{+} here. □_{10.5}

∗  ∗  ∗

So we have finally proved the main theorem. Though not directly contributing to our main theme, we remark some more consequences of ≤_{𝔎}↾ K_{λ^{+}} ≠≤_{λ^{+}}^{∗}.

10.6 Claim. (∗)_{0} ⇔ (∗)_{1} where

(∗)_{0} for some M ≤_{𝔎} N from K_{λ^{+}}, we do not have M ≤_{λ^{+}}^{∗} N (∗)_{1} for some M ≤_{𝔎} N from K_{λ^{+}} we have:

if a ∈ N\M \underline{then} tp(a, M, N) is not realized in any M

such that M^{∗} ≤_{λ^{+}}^{∗} M ∈ K_{λ^{+}}

10.7 Definition. Assume M_{0} <_{𝔎} M_{1} are in K_{λ^{+}},and M̄^{ℓ} = ⟨M_{ℓ,i} : i < λ^{+}⟩ is a ≤_{𝔎}-representation of M_{ℓ} for ℓ = 0, 1. Let

(a)  𝒮_{0}(M̄^{0}, M̄^{1}) = {δ < λ^{+} : M_{1,δ} ∩ M_{0} = M_{0,δ}
and not NF_{λ}(M_{0,δ}, M_{0,δ+1}, M_{1,δ}, M_{1})}

(b)  𝒮_{1}(M_{0}, M_{1}) = 𝒮_{0}(M̄^{0}, M̄^{1})/D_{λ^{+}} (well defined).

(c) J is the normal ideal on λ^{++} generated by sets of the form 𝒮_{0}(M^{0}, M^{1}) where M^{0}, M^{1} are as above.

10.8\underline{Comment}: An earlier try for 10.4 was:

1) For every S ∈ J for some M̄^{0}, M^{1} as in 10.7 we have S = S_{0}(M̄^{0}, M̄^{1}).

2) If S_{1} = S_{1}(M^{0}, M^{1}) is stationary then for some M̄ = ⟨M_{i} : i < λ^{+}⟩ a representation of M = ⋃_{i<λ^{+}} M_{i} ∈ K_{λ^{+}} for every S ⊆ S_{1} for some M̄′, we have

M̄^{1} = ⟨M_{i}^{1} : i < λ^{+}⟩, M_{1} <_{𝔎} M^{1} = ⋃_{i<λ^{+}} M_{i}^{1}, M_{i}^{1} ∩ M = M_{i} and

S_{0}(M̄, M̄^{1}) = S mod 𝒟_{λ^{+}}.

3) If ≤_{λ^{+}}^{∗}≠≤_{𝔎}↾ K_{λ^{+}} and I(λ^{+2}, K) < 2^{λ^{+2}} \underline{then} for some stationary S ⊆ λ^{+} we have:

(a) 𝒟_{λ^{+}} ↾ S is λ^{++}-saturated

(b) M̄^{0}, M̄^{1} as in 10.4 implies S_{1}(M̄^{0}, M̄^{1}) ⊆ S mod D_{λ^{+}}.

4) If ≤_{λ^{+}}^{∗}≠≤_{𝔎}↾ K_{λ^{+}} and 2^{λ} < 2^{λ^{+}} < 2^{λ^{+2}} then I(λ^{+2}, K) = 2^{λ^{+2}}.

Proof. 1) First we prove only “S ⊆ S_{0}(M̄^{0}, M̄^{1})”. Easy as 𝔎_{λ^{+}} has amalgamation and

⨂ if M_{0} ≤_{𝔎} M_{1} ≤_{𝔎} M_{2} are in K_{λ^{+}}, M̄^{ℓ}
representing the S_{0}(M̄^{0}, M̄^{1}) ⊆ S_{0}(M̄^{0}, M̄^{2}).

Now for equality use part (2).

2) Similar to the proof of 10.2.

3) Suppose S^{∗} = S_{1}(M̄^{0}, M̄^{2}) is stationary, let S̄ = ⟨S′_{α} : α < λ^{++}⟩ be such that S′_{α} ∈ J. We can build a model M S̄ ∈ K_{λ^{+2}} and a representation ⟨M S̄ ; α : α < λ^{++}⟩ such that

S_{1}(M_{α}, M_{α+1}) = S′_{α}/D_{λ^{++}}.

4) By §3 (using the proof of part (2)).

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