Place: Fields Institute (Room 210)

Date: March 24, 2017 (13:30-15:00)

Speaker: Zoltán Vidnyánszky, York University

Title: Anti-basis results for graphs of infinite Borel chromatic number

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.