Georgia Institute of Technology College of Sciences

We discuss the relationship between the Borel/Baire measurable/measurable combinatorics of the action of a finitely generated group on its Bernoulli shift and the discrete combinatorics of the multiplication action of that group on itself. Our focus is on various chromatic numbers of graphs generated by these actions. We show that marked groups with isomorphic Cayley graphs can have Borel/Baire measurable/measurable chromatic numbers which differ by arbitrarily much. In the Borel two-ended, Baire measurable, and measurable hyperfinite settings, we show our constructions are nearly best possible (up to only a single additional color). Along the way, we get tightness of some bounds of Conley and Miller on Baire measurable and measurable chromatic numbers of locally finite Borel graphs.