# 2017 seminar talk: Borel chromatic numbers: finite vs infinite

Talk held by Zoltán Vidnyánszky (York University, Toronto, Canada and Toronto University) at the KGRC seminar on 2017-06-08.

### Abstract

One of the most interesting results of Borel graph combinatorics is the $G_0$ dichotomy, i. e., the fact that a Borel graph has uncountable Borel chromatic number if and only if it contains a Borel homomorphic image of a graph called $G_0$. It was conjectured that an analogous statement could be true for graphs with infinite Borel chromatic number. Using descriptive set theoretic methods we answer this question and a couple of similar questions negatively, showing that one cannot hope for the existence of a Borel graph whose embeddability would characterize Borel (or even closed) graphs with infinite Borel chromatic number. We will also discuss a positive result and its relation to Hedetniemi's conjecture.