A bound on measurable chromatic numbers of locally finite Borel graphs

A graph on a set X is an irreflexive, symmetric set G ⊆ X ×X. Such a graph is locally finite if every point has only finitely many G-neighbors. A (κ-)coloring of such a graph is a function c : X → κ with the property that ∀(x, y) ∈ G c(x) = c(y). The chromatic number of such a graph, or χ(G), is the least cardinal κ for which there is such a κ-coloring. Note that any locally finite graph may be colored with countably many colors. In this paper, we consider measurable analogs of these notions, a subject of increasing interest over the last few years due to its connections with descriptive set-theoretic dichotomies and dynamical properties of group actions. A subset of a topological space is Borel if it is in the σ-algebra generated by the underlying topology, and a function between topological spaces is Borel if pre-images of open sets are Borel. A Polish space is a separable topological space which admits a compatible complete metric. While it is hardly standard terminology, we use the term Polish cardinal to refer to a cardinal equipped with a Polish topology. Thus the Polish cardinals are exactly those in the set {0, 1, . . . ,א0, 2א0}, with the two infinite cardinals supporting various topologies.