LCLs in the Borel Hierarchy

TL;DR

This paper investigates the complexity of solutions to locally checkable labeling problems (LCLs) on groups within the Borel hierarchy, showing that for certain LCLs, solutions exist at higher Baire classes but not at lower ones, revealing a nuanced hierarchy.

Contribution

It constructs specific LCLs on free groups that demonstrate a strict hierarchy of Baire class solutions, advancing understanding of descriptive combinatorics in the Borel hierarchy.

Findings

Existence of LCLs with solutions only at higher Baire classes

Demonstration of a strict hierarchy in Baire class solutions

Insights into the gap between Borel and continuous solutions

Abstract

A locally checkable labeling problem (LCL) on a group $\Gamma$ asks one to find a labeling of the Cayley graph of $\Gamma$ satisfying a fixed, finite set of "local" constraints. Typical examples include proper coloring and perfect matching problems. In descriptive combinatorics, one often considers the existence of solutions to LCLs in the setting of descriptive set theory. For example, given a free action of $\Gamma$ on a Polish space $X$, we might be interested in solving a given LCL on each orbit in a continuous, Borel, measurable, etc. way. In an attempt to understand more finely the gap between Borel and continuous combinatorics, we consider the existence of Baire class $m$ solutions to LCLs. For all $n > 1$ and $m \in \omega$, we produce an LCL on $\mathbb{F}_n$ which always admits Baire class $m+1$ solutions, but not necessarily Baire class $m$ solutions.