Marked Groups with Isomorphic Cayley Graphs but Different Borel Combinatorics

TL;DR

This paper constructs pairs of marked groups with identical Cayley graphs but differing Borel and measure chromatic numbers, addressing open questions in descriptive combinatorics and group theory.

Contribution

It provides explicit examples of groups with isomorphic Cayley graphs yet different Borel and measure chromatic properties, solving longstanding open problems.

Findings

Constructed pairs of groups with isomorphic Cayley graphs but different Borel chromatic numbers.

Showed these graphs have distinct Baire measurable and measure chromatic numbers.

Answered open questions posed by Kechris and Marks regarding Borel combinatorics.

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.