LCLs in the Borel Hierarchy

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.

• The paper demonstrates the stratification between continuous and Borel solvability for Labelable ternary free labels on a free group that admits Baire class solutions for one level, which does not admit them for class $m$
• The paper proves that any continuous solution of a labeling problem leads to lower-hard classes in the hierarchy off the hair if the free part of Bernoulli shift satisfies separated into disjoint.
• To identify the lower order complexity on free groups by finding Borel invariant solutions, techniques take from cartesian category, Mates’ theorem, and uniform determinability across defined classes.

This paper, "LCLs in the Borel Hierarchy" (2601.19046), by Felix Weilacher, studies locally checkable labeling problems (LCLs) on finitely generated groups from the standpoint of descriptive set theory. Its main contribution is a strict stratification of the gap between continuous and Borel solvability: for every finite level of the Baire hierarchy there is an LCL on the free group $F_2$ that admits Baire class $m+1$ solutions on every free action but not necessarily Baire class $m$ solutions.

Background and complexity classes

An LCL on a countable group $\Gamma$ is a triple $\Pi = (\Lambda, S, \mathcal{A})$ consisting of a finite label set $\Lambda$, a finite window $S \subseteq \Gamma$, and a constraint family $\mathcal{A} \subseteq \Lambda^S$. A solution labels each point $x$ of a free action $\Gamma \curvearrowright X$ so that the pattern of labels on $m+1$0 lies in $m+1$1; proper coloring and perfect matching are standard examples. The paper organizes LCLs into complexity classes: $m+1$2, $m+1$3, $m+1$4, $m+1$5, and $m+1$6, according to which free actions admit solutions of the corresponding regularity.

For nonabelian free groups, prior work in (Brandt et al., 2021), combining results of Kechris–Solecki–Todorcevic, Conley–Marks–Tucker-Drob, Marks, and others, established that all these inclusions are strict for $m+1$7, $m+1$8, with witness problems such as $m+1$9-coloring in $m$0 and 3-coloring in $m$1. The paper refines the interval between $m$2 and $m$3 using the Baire hierarchy: an LCL is in $m$4 if every free continuous action on a 0-dimensional Polish space admits a Baire class $m$5 solution. Since label sets are finite, this amounts to requiring preimages of labels to be $m$6.

Universality via the Bernoulli shift

The paper establishes that the free part $m$7 of the Bernoulli shift is universal: any free continuous action on a 0-dimensional Polish space embeds continuously equivariantly into it, and any free Borel action embeds Borel-equivariantly. Consequently, $m$8 if and only if $m$9 admits a Borel $\Gamma$0-labeling, and similarly at each Baire class $\Gamma$1. This yields the identity

$\Gamma$2

so the hierarchy cannot strictly increase through all countable ordinals, and the question of where it stabilizes is well posed. Subgroup transfer also shows all results for $\Gamma$3 apply to $\Gamma$4, since each $\Gamma$5 embeds in $\Gamma$6.

Hardness of 2-coloring orbits

A key technical tool concerns proper 2-coloring of $\Gamma$7-actions with respect to the generator $\Gamma$8, which is not Borel solvable [kst]. Letting $\Gamma$9 be the $\Pi = (\Lambda, S, \mathcal{A})$0-ideal of $\Pi = (\Lambda, S, \mathcal{A})$1-invariant Borel subsets of $\Pi = (\Lambda, S, \mathcal{A})$2 admitting Borel 2-colorings, the paper proves that members of $\Pi = (\Lambda, S, \mathcal{A})$3 are meager in every compatible Polish topology, and then proves a stronger statement: for each $\Pi = (\Lambda, S, \mathcal{A})$4 there is a $\Pi = (\Lambda, S, \mathcal{A})$5-complete set $\Pi = (\Lambda, S, \mathcal{A})$6 such that $\Pi = (\Lambda, S, \mathcal{A})$7 is $\Pi = (\Lambda, S, \mathcal{A})$8-hard for every $\Pi = (\Lambda, S, \mathcal{A})$9. The proof uses Máté's topological Hurewicz test pairs [matrai2007covering]: canonical sets $\Lambda$0 that are closed nowhere dense in refined topologies $\Lambda$1, together with the category criterion that $\Lambda$2 sets nonmeager in $\Lambda$3 remain nonmeager in $\Lambda$4. Combined with Wadge determinacy, this converts category arguments into Borel-hierarchy lower bounds. This theorem is of independent interest as a mechanism for proving lower bounds on definable colorings.

The main theorem and its proof

The main result is:

Theorem. For each $\Lambda$5, $\Lambda$6.

More precisely, for each $\Lambda$7 the paper constructs an LCL $\Lambda$8 in $\Lambda$9 whose label set contains a distinguished subset $S \subseteq \Gamma$0 such that in any Borel $S \subseteq \Gamma$1-labeling, the preimage of $S \subseteq \Gamma$2 is $S \subseteq \Gamma$3-hard — which immediately rules out Baire class $S \subseteq \Gamma$4 solutions.

The construction follows a uniform scheme. The second coordinate of the labeling solves some auxiliary problem on $S \subseteq \Gamma$5-orbits; points whose second coordinate falls in a designated set must carry first coordinate $S \subseteq \Gamma$6; first coordinate $S \subseteq \Gamma$7 labels are required to be $S \subseteq \Gamma$8-invariant along a new $S \subseteq \Gamma$9 direction generated by $\mathcal{A} \subseteq \Lambda^S$0; and the labels $\mathcal{A} \subseteq \Lambda^S$1 give a partial 2-coloring of each $\mathcal{A} \subseteq \Lambda^S$2-orbit. The design principle is that entirely "good" $\mathcal{A} \subseteq \Lambda^S$3-orbits are strongly encouraged to use the $\mathcal{A} \subseteq \Lambda^S$4 label, since the alternative — full 2-coloring — is categorically impossible to perform Borelly on invariant sets, while orbits containing a $\mathcal{A} \subseteq \Lambda^S$5 point can be 2-colored easily by counting distance to the nearest $\mathcal{A} \subseteq \Lambda^S$6.

In the base case ($\mathcal{A} \subseteq \Lambda^S$7), the auxiliary problem is continuous 3-coloring of $\mathcal{A} \subseteq \Lambda^S$8-orbits, available by [kst, bernshteyn2023distributed]. Nonemptiness of the $\mathcal{A} \subseteq \Lambda^S$9-labeled set in any Borel solution relies on Marks' lemma [marks2017uniformity], a Borel determinacy result producing continuous equivariant maps into or avoiding a given Borel set: if no orbit were labeled $x$0, both alternatives would yield forbidden Borel 2-colorings of $x$1. Empty interior of the $x$2-set follows from generic ergodicity of the $x$3-action together with $x$4-independence of the green set. Notably, the same LCL placed on $x$5 would admit solutions with empty $x$6-set, since $x$7 has a Borel 3-coloring [GJKS_borel]; the hardness genuinely exploits nonamenability of $x$8.

The inductive step applies the same scheme over $x$9: assuming $\Gamma \curvearrowright X$0 with $\Gamma \curvearrowright X$1 giving $\Gamma \curvearrowright X$2-hardness for $\Gamma \curvearrowright X$3, the lifted problem $\Gamma \curvearrowright X$4 lies in $\Gamma \curvearrowright X$5 and its $\Gamma \curvearrowright X$6-preimage is $\Gamma \curvearrowright X$7-hard. Here Marks' lemma is applied to a set defined using the winning condition of the Wadge game, effectively composing the two determinacy arguments, and Theorem on 2-coloring complexity supplies the final hardness bound. By subgroup transfer, the separation holds for $\Gamma \curvearrowright X$8 itself.

Limitations and open questions

The result covers only finite levels of the Baire hierarchy, and the paper is explicit that it cannot extend indefinitely: since there are only countably many LCLs and the Bernoulli shift is universal, $\Gamma \curvearrowright X$9. The author conjectures stabilization already at $m+1$00, i.e., that $m+1$01, but this remains open. For abelian groups $m+1$02, $m+1$03, proper 3-coloring is known to lie in $m+1$04 but not $m+1$05, and whether $m+1$06 is unresolved. More broadly, the paper asks whether some countable group pushes the hierarchy past $m+1$07, or arbitrarily far below $m+1$08, and whether projective predicates can be encoded by local problems under Projective Determinacy — a direction the author notes is entirely unexplored.

Conclusion

The paper demonstrates that the descriptive complexity of LCLs on free groups is sensitive to every finite level of the Baire hierarchy, providing the first systematic study of Baire-class solvability above continuity within the LCL framework. Its combination of Máté's category-based lower-bound machinery with Marks' determinacy lemma offers a reusable template for separating definability classes, and the sharp dependence on nonamenability — the same construction collapses on $m+1$09 — delineates where these methods apply.