Logic Seminar

We consider the problem of finding a 2-regular subgraph of a given regular graph with no odd cycles. We show that this is always possible in the BP context. As a consequence, odd-regular bipartite Borel graphs on Polish spaces admit perfect matchings with the property of Baire, in contrast with recent examples of Kun in the measure-theoretic setting. Analogous results in the measure-theoretic context hold for hyperfinite graphs. This is joint work with Matt Bowen and Felix Weilacher, building upon prior joint work with Kechris and with Miller.