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Abstract. Ultraproducts of measure preserving actions of countable groups are used to study the graph combinatorics associated with such actions, including chromatic, independence and matching numbers. Applications are also given to the theory of random colorings of Cayley graphs and sofic actions and equivalence relations.

We study in this paper combinatorial problems concerning graphs generated by measure preserving actions of countable groups on standard measure spaces. In particular we study chromatic and independence numbers, in both the measure-theoretic and the Borel context, and relate the behavior of these parameters to properties of the acting group such as amenability, Kazhdan's property (T), and freeness. We also prove a Borel analog of the classical Brooks' Theorem in finite combinatorics for actions of groups with finitely many ends.

Abstract. We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes") with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz-HuczekZhang tiling theorem for countable amenable groups and strengthens the Ornstein-Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-stable.