Tight degeneracy bounds in online Ramsey games

Abstract

In the $q$-color online Ramsey game, Builder and Painter play on an infinite independent set of vertices. At each step, Builder draws an edge and Painter immediately assigns it one of $q$ colors. Builder aims to force a monochromatic copy of a fixed graph $H$. We prove that, for every $q \ge 2$ and $d \ge 1$, Builder can force a monochromatic copy of any $d$-degenerate graph $H$ while drawing a graph of degeneracy at most $d$. The bound $d$ is tight, and this resolves in the affirmative a problem of Conlon, Fox and Sudakov.