Marked Groups with Isomorphic Cayley Graphs but Different Borel Combinatorics (1810.03062v2)

Abstract: We construct pairs of marked groups with isomorphic Cayley graphs but different Borel chromatic numbers for the free parts of their shift graphs. This answers a question of Kechris and Marks. We also show that these graphs have different Baire measurable and measure chromatic numbers, answering analogous versions of the question.