Natasha Dobrinen : Ramsey theory on binary relational homogeneous structures

Posted by Tom Benhamou , part of the Logic Seminar.

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April 11, 2023, 3:30 p.m.

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Abstract

Generalizations of Ramsey's Theorem to colorings of infinite sets proceeds via topological considerations. The Galvin-Prikry Theorem states that Borel subsets of the Baire space are Ramsey. Silver extended this to analytic sets, and Ellentuck gave a topological characterization of Ramsey sets in terms of the property of Baire in the Vietoris topology. We extend these theorems to several classes of countable homogeneous structures, answering a question of Kechris, Pestov, and Todorcevic. An obstruction to exact analogues of Galvin-Prikry or Ellentuck is the presence of big Ramsey degrees. We will discuss how different properties of the structures affect which analogues have been proved. Presented is work of the speaker for SDAP+ structures, and joint work with Zucker for binary finitely constrained FAP classes. In both works, the pigeonhole principle is achieved in ZFC by repeated applications of the forcing mechanism to find a finite object with desired properties. A feature of the work with Zucker is showing that we can weaken one of Todorcevic's four axioms guaranteeing a Ramsey space, and still achieve the same conclusion. These axioms are built on prior work of Carlson and Simpson developing topological Ramsey space theory.