We will start with stating two concrete Ramsey results of the type we are interested in: Gowers' tetris theorem and Furstenberg and Katznelson's generalization of the Hales-Jewett theorem. We will describe a general framework for such results in terms of partial semigroups. We will then introduce new algebraic structures appropriate for formalizing such theorems: this will involve in a crucial way ultrafilter spaces over partial semigroups. We will phrase and prove a general Ramsey theorem on the existence of appropriately defined basic sequences for such structures. We will proceed to describing concrete examples of such algebraic structures, which will involve monoid actions by endomorphisms on compact left-topological semigroups and operations of tensor product and substructure. As applications, we will derive the Gowers and Furstenberg-Katznelson theorems. Time permitting, we will finish with some further applications.
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