Borel Combinatorics of Schreier Graphs of $\mathbb{Z}$-actions

TL;DR

This paper explores the Borel combinatorics of Schreier graphs from $bZ$-actions, establishing decidability and algorithms for Borel chromatic numbers, contrasting with more complex $bZ^2$-actions.

Contribution

It introduces a framework linking Borel combinatorics to subshifts of finite type and provides an exponential-time algorithm for computing Borel chromatic numbers.

Findings

Borel and continuous combinatorics coincide for $bZ$-actions.

The Borel chromatic number problem is decidable and solvable in exponential time.

A formula for the Borel chromatic number when the generating set size is 4.

Abstract

In this paper we consider the Borel combinatorics of Schreier graphs of $\mathbb{Z}$-actions with arbitrary finite generating sets. We formulate the Borel combinatorics in terms of existence of Borel equivariant maps from $F(2^{\mathbb{Z}})$ to subshifts of finite type. We then show that the Borel combinatorics and the continuous combinatorics coincide, and both are decidable. This is in contrast with the case of $\mathbb{Z}^2$-actions. We then turn to the problem of computing Borel chromatic numbers for such graphs. We give an algorithm for this problem which runs in exponential time. We then prove some bounds for the Borel chromatic numbers and give a formula for the case where the generating set has size 4.