In this article we survey the emerging field of descriptive graph combinatorics. This area has
developed in the last two decades or so at the interface of descriptive set theory and graph …

We show that there is no simple (eg finite or countable) basis for Borel graphs with infinite
Borel chromatic number. In fact, it is proved that the closed subgraphs of the shift graph on …

We first consider the class K of graphs on a zero-dimensional metrizable compact space
with continuous chromatic number at least three. We provide a concrete basis of size …