University of California Irvine

In this talk, we discuss existentially closed measure preserving actions of countable groups.  A classical result of Berenstein and Henson shows that the model companion for this class exists for the group of integers and their analysis readily extends to cover all amenable groups.  Outside of the class of amenable groups, relatively little was known until recently, when Berenstein, Henson, and Ibarlucía proved the existence of the model companion for the case of finitely generated free groups.  Their proof relies on techniques from stability theory and is particular to the case of free groups.  In this talk, we will discuss the existence of model companions for measure preserving actions for the much larger class of universally free groups (also known as fully residually free groups), that is, groups which model the universal theory of the free group.  We also give concrete axioms for the subclass of elementarily free groups, that is, those groups with the same first-order theory as the free group.  Our techniques are ergodic-theoretic and rely on the notion of a definable cocycle.  This talk represents ongoing work with Brandon Seward and Robin Tucker-Drob.

We present an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two.
This is a joint work with David Conlon.