Benjamin Miller : "Applications of the open graph dichotomy"

The open graph dichotomy is a generalization of the perfect set theorem, ensuring that every open graph on an analytic set has either a countable coloring or a perfect clique. As the proof of this result is essentially the same as that of the perfect set theorem, it can be viewed as one of the very simplest descriptive set-theoretic dichotomy theorems. Nevertheless, there is an infinite-dimensional analog of the open graph dichotomy (whose proof is essentially the same) that has recently proven particularly useful in studying Borel functions, graphs, and sets of low complexity.

We will begin by stating and proving the infinite-dimensional analog of the open graph dichotomy. We will then describe how it can be used to give particularly simple proofs of several well-known facts, such as the Hurewicz dichotomies, the Jayne-Rogers theorem, and Lecomte’s characterization of the existence of countable Borel colorings of low complexity. Finally, we will turn our attention to the new result that there is a twenty-four element basis, under closed continuous embeddability, for the class of Borel functions that are not Baire class one.

• Beginning: Kechris "Classical descriptive set theory," first three chapters.
• Advanced:
  • Jayne and Rogers, "First level Borel functions and isomorphisms", Journal de Mathématiques Pures et Appliquées 61 (1982), 177–205.
  • Solecki "Decomposing Borel sets and functions and the structure of Baire class 1 functions", Journal of the American Mathematical Sociey 11 (1998), 521–550.
  • Lecomte and Zeleny, "Baire class ξ colorings: the first three levels" Transactions of the American Mathematical Society 366 (2014), 2345–2373.

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