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.
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Clinton T. Conley, Alexander S. Kechris, Measurable chromatic and independence numbers for ergodic graphs and group actions. Groups Geom. Dyn. 7 (2013), no. 1, pp. 127–180