Abstract:

I will discuss some recent developments in canonical Ramsey theory in the context of descriptive set theory. The talk will be accessible to non-specialists.
Given a class E of Borel equivalence relations and a sigma-ideal I on a Polish space X, we say that the class canonizes to a finite set {F₁, ... ,F_n} of equivalence relations if every Borel equivalence relation in the class E is equal to one of F₁, ... ,F_n after a restriction to a Borel set not in I. I will survey recent canonization results for various classes of Borel equivalence relations.
This is joint work with Jindra Zapletal and Vladimir Kanovei.