Definable combinatorics in hyperfinite graphs

Combinatorics Seminar
Friday, March 18, 2022 - 3:00pm for 1 hour (actually 50 minutes)
Skiles 005
Matthew Bowen – McGill University – matthew.bowen2@mail.mcgill.ca
Anton Bernshteyn

We discuss a few new results concerning the descriptive combinatorics of bounded degree hyperfinite Borel graphs. In particular, we show that the Baire measurable edge chromatic number of $G$ is at most $\lceil\frac{3}{2}\Delta(G)\rceil+6$ when G is a multigraph, and for bipartite graphs we improve this bound to $\Delta(G)+1$ and show that degree regular one-ended bipartite graphs have Borel perfect matchings generically. Similar results hold in the measure setting assuming some hyperfiniteness conditions. This talk is based on joint work with Kun and Sabok, Weilacher, and upcoming work with Poulin and Zomback.