A central goal in the area of countable Borel equivalence relations (CBERs) is understanding hyperfinite CBERs. So far, roughly speaking, all known proofs for non-hyperfiniteness use measures, while the tool of Baire category is not useful in this context, as all CBERs are hyperfinite on a co-meager invariant set. Regarding topological Ramsey spaces (in the sense of Todorčević, a classical result of Mathias and Soare states that every CBER on the Ellentuck space \([{\mathbb N}]^{\mathbb N}\) is hyperfinite on a Ramsey positive set. There are similar results for certain other topological Ramsey spaces: Kanovei, Sabok and Zapletal proved the analogous canonization result for the Milliken space, and recently, Panagiotopulos and Wang showed that CBERs are hyperfinite (in fact, even smooth) on positive sets in the Carlson-Simpson space. We generalize these statements to all topological Ramsey spaces.

This is joint work with Zoltán Vidnyánszky.