We define games which characterize countable coloring numbers of analytic graphs on Polish spaces. These games can provide simple verification of the countable chromatic number of certain graphs. We also get a simpler proof of a dichotomy originally proved by Adams and Zapletal: If an analytic graph has an uncountable coloring number, then it contains the graph \(\Delta_0\) as a subgraph. (Here the graph \(\Delta_0\) is a certain simple graph with uncountable coloring number.)
Joint work with Jindrich Zapletal.