Toronto Set Theory

Event Information Anti-basis results for graphs of infinite Borel chromatic number
13:30 on Friday March 24, 2017
15:00 on Friday March 24, 2017
FI210, Fields Institute, 222 College St.
Zoltan Vidnyánszky
http://www.yorku.ca/vidnyanz/
York University
http://mathstats.info.yorku.ca/

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.