On Baire Measurable Colorings of Group Actions

The field of descriptive combinatorics investigates the question, to what extent can classical combinatorial results and techniques be made topologically or measure-theoretically well-behaved? This paper examines a class of coloring problems induced by actions of countable groups on Polish spaces, with the requirement that the desired coloring be Baire measurable. We show that the set of all such coloring problems that admit a Baire measurable solution for a particular free action $\alpha$ is complete analytic (apart from the trivial situation when the orbit equivalence relation induced by $\alpha$ is smooth on a comeager set); this result confirms the "hardness" of finding a topologically well-behaved coloring. When $\alpha$ is the shift action, we characterize the class of problems for which $\alpha$ has a Baire measurable coloring in purely combinatorial terms; it turns out that closely related concepts have already been studied in graph theory with no relation to descriptive set theory. We remark that our framework permits a wholly dynamical interpretation (with colorings corresponding to equivariant maps to a given subshift), so this article can also be viewed as a contribution to generic dynamics.