Logic Seminar

One of the most interesting results of Borel graph combinatorics is the $G_0$ dichotomy, i. e., the fact that an analytic 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 of infinite Borel chromatic number, or at least for some well-behaving subclass of the class of infinitely chromatic graphs. Using descriptive set theoretic methods we produce examples showing that some versions of these statements are false.