I would like to give an overview of recent results in canonical Ramsey theory in the context of descriptive set theory. This is the subject of a recent monograph joint with with Vladimir Kanovei and Jindra Zapletal. The main question we address is the following. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? Canonical Ramsey theory stems from finite combinatorics and is concerned with finding canonical forms of equivalence relations on finite (or countable) sets. We obtain canonization results for analytic and Borel equivalence relations and in cases when canonization is impossible, we prove ergodicity theorems. For a publisher’s book description see:

http://www.youtube.com/watch?v=ZPXBcKeepKk

We will discuss a question asked by Stefan Geschke, whether the existence of a selector for the equivalence relation $E_0$ implies the existence of a nonprincipal ultrafilter on the integers. We will present a negative solution which is undoubtedly more complicated than necessary, using a variation of Woodin’s $mathbb{P}_{mathrm{max}}$. This proof shows that, under suitable hypotheses, if $E$ is a universally Baire equivalence relation on the reals, with countable classes, then forcing over $L(E,mathbb{R})$ to add a selector for $E$ does not add a nonprincipal ultrafilter on the integers.