LCLs in the Borel Hierarchy

TL;DR

This work studies the definability of solutions to locally checkable labeling problems (LCLs) on group actions within descriptive set theory, focusing on the BAIRE hierarchy between Borel and continuous settings. The author constructs a universal Bernoulli shift framework and proves that for the free group $\mathbb{F}_2$ the BAIRE hierarchy is strictly increasing at every finite level: $\mathtt{BAIRE}_n(\mathbb{F}_2) \subsetneq \mathtt{BAIRE}_{n+1}(\mathbb{F}_2)$ for all $n \in \omega$, and shows how this separation propagates to all non-abelian free groups via subgroup embeddings. The key technical contributions include a base case LCL on $\mathbb{F}_2$ that encodes a $\mathbf{\Pi}^0_1$-hard set and an inductive step that lifts complexity from $\Gamma$ to $\mathbb{Z} * \Gamma$ by combining a carefully designed coordinate encoding with a $\mathbf{\Sigma}^0_{1+\alpha}$-complete witness, aided by Marks’ lemma and topological-Hurewicz machinery. The paper ends with open questions about the exact point where BAIRE equals BOREL and potential extensions to other groups and to projective predicates under determinacy. These results illuminate the fine-grained structure of definable LCLs and establish a robust framework for hierarchy-based separations in descriptive combinatorics.

Abstract

A locally checkable labeling problem (LCL) on a group $Γ$ asks one to find a labeling of the Cayley graph of $Γ$ 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 $Γ$ 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 $m \in ω$, we produce an LCL on $\mathbb{F}_n$ which always admits Baire class $m+1$ solutions, but not necessarily Baire class $m$ solutions.

Figures (3)

• Figure 1: The complexity classes introduced so far for $\mathbb{F}_n$, $n > 1$. Arrows between classes denote inclusion, and blue arrows indicate that the inclusion is strict. To the left of each class is an LCL belonging to that class but no lower one. Each $k$-coloring problem refers to the Schreier graphs generated by the usual generating set for $\mathbb{F}_n$.
• Figure 2: A part of a Schreier graph of $\mathbb{F}_2$ with a $\Pi$-labeling in the base case $n = 1$. The $a$ and $b$ orbits are drawn horizontaly and vertically respectively. The second coordinates of each label are shown as the color of a vertex while the first coordinates are simply written on the vertex.
• Figure 3: The three ways a $\mathbb{Z}$-orbit can look after pulling back the $\Pi'$-labeling via $f$. Points are colored red if and only if they are labeled with $*$ in order to maintain the analogy with the base case and Figure \ref{['fig:base_case']}.

Theorems & Definitions (37)

• Definition 1.1: NaorStock
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Proposition 1.6
• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3