Borel Distinguishing Number

• Borel distinguishing number is a variant requiring a Borel measurable coloring to break all non-identity graph automorphisms.
• It exposes dramatic separation phenomena, where graphs with low classical distinguishing numbers may need uncountably many colors when measurability is enforced.
• Methodologies such as Borel transversals, trajectory analysis, and entropy arguments underscore the gap between abstract and Borel symmetry breaking.

The Borel distinguishing number is a definable variant of the classical distinguishing number of a graph, recast in the descriptive set theoretic context. This concept measures the minimal cardinality of a color set needed for a Borel measurable coloring that breaks all graph automorphisms—that is, it distinguishes every non-identity automorphism by ensuring at least one vertex moves to a different color. The paper of Borel distinguishing numbers reveals that embedding the coloring problem into the framework of Borel graphs and measurable automorphisms yields fundamentally new phenomena and separation results not visible in the purely combinatorial setting.

1. Definitions: Classical and Borel Distinguishing Numbers

Let $G = (X, E)$ be a graph where $X$ is the vertex set and $E \subset X \times X$ is the edge relation. A distinguishing coloring is any map $c : X \to I$ (with $I$ an indexing set) for which every $\varphi \in \operatorname{Aut}(G)$, $\varphi \neq \mathrm{id}$, fails to preserve all color classes: there exists $x$ with $c(x) \neq c(\varphi(x))$. The classic distinguishing number is

$$D(G) = \min \{\,|I| : \text{there exists a distinguishing coloring } c : X \to I\,\}.$$

In the Borel setting, $X$ is a standard Borel space and $E$ is a Borel subset of $X\times X$. A Borel distinguishing coloring is a Borel measurable map $c : X \to I$ distinguishing all automorphisms, while a strictly Borel distinguishing coloring requires only distinguishing Borel automorphisms, i.e., those in the subgroup $\operatorname{Aut}_B(G)$. Denote these associated invariants as

$$D_B(G) = \min \{\,|I| : \text{Borel distinguishing coloring of } G \},$$

$$D_{SB}(G) = \min \{\,|I| : \text{strictly Borel distinguishing coloring of } G \}.$$

By construction,

$D(G) \leq D_B(G),\quad D_{SB}(G) \leq D_B(G).$

For graphs where $\operatorname{Aut}_B(G)$ is trivial but $\operatorname{Aut}(G)$ is not, one can have $D_{SB}(G) = 1 < D(G)$.

2. Main Theorems: Separation Phenomena

The paper (Bilge et al., 11 Aug 2025) establishes that the Borel distinguishing number can differ sharply from $D(G)$; separation results are demonstrated at multiple levels:

• Countable Separation: For locally countable Borel graphs with smooth connectedness relation, a Borel coloring exists using at most countably many colors:

$D_B(G), D_{SB}(G) \leq \aleph_0$

(Theorem 3.1), achieved by selecting a Borel transversal for the smooth equivalence relation and coloring connected components uniquely via a Borel injection into $\mathcal{P}(\mathbb{N})$.

• Uncountable Separation: For Borel graphs arising from nonsmooth countable Borel equivalence relations, every abstract connected component is a countable clique ($K_\mathbb{N}$), thus $D(G) = \aleph_0$. However, by Feldman–Moore's theorem, no Borel transversal exists for nonsmooth relations, so any Borel distinguishing coloring requires continuum-sized color set:

$D_B(G) = D_{SB}(G) = 2^{\aleph_0}$

(Theorem 4.1).

• Separation Below $\aleph_0$: For every integer $n \geq 3$ there exist Borel graphs (e.g., the free part of the shift graph $\mathbb{Z} \curvearrowright n^\mathbb{Z}$) with $D(G) = 2$ but requiring at least $n$ colors for a Borel distinguishing coloring:

$D_B(G), D_{SB}(G) \geq n$

The construction involves encoding each vertex’s color trajectory using a Borel-equivariant map $\mathrm{tr}_f: X \rightarrow k^\mathbb{Z}$ and entropy arguments to show injectivity is impossible when $k < n$; optimal $k = 2n-1$ is constructed in the paper.

3. Conceptual and Methodological Framework

• Automorphism Groups: The classical invariant $D(G)$ ignores definability; all symmetries of $G$ are considered. In the Borel regime, focus shifts to those automorphisms that are themselves Borel, or to all automorphisms but only Borel colorings.
• Definability Constraints: Borel coloring requires the coloring function to be Borel measurable. This restriction is often severe for infinite graphs, as exhibited by the failure of Borel transversals in nonsmooth equivalence relations.
• Trajectory Techniques: In shift graphs, the coloring is analyzed via the trajectory map $tr_f$ capturing color sequences along group actions, with injectivity conditions ensuring distinguishability. Entropy and combinatorial constraints establish lower bounds for the color set.

4. Illustrative Examples and Analytical Results

The following table illustrates separation phenomena:

Graph/Class	$D(G)$	$D_B(G)$	$D_{SB}(G)$
Locally countable, smooth Borel graph	$\leq \aleph_0$	$\leq \aleph_0$	$\leq \aleph_0$
Nonsmooth Borel equivalence relation ($K_\mathbb{N}$ components)	$\aleph_0$	$2^{\aleph_0}$	$2^{\aleph_0}$
Shift graph $\mathbb{Z} \curvearrowright n^\mathbb{Z}$, $n\geq3$	$2$	$\geq n$, $k=2n-1$ can be achieved	$\geq n$, $k=2n-1$ can be achieved

These results demonstrate that abstract low distinguishing numbers do not guarantee corresponding low Borel distinguishing numbers; definability constraints can force a large increase.

5. Implications and Open Questions

The Borel distinguishing number signals a strong “definability cost” in symmetry breaking for infinite graphs. This phenomenon is especially pronounced in the presence of nonsmooth Borel equivalence relations, where the absence of Borel transversals inflates the Borel coloring requirements to continuum cardinality. Even for shift graphs and other regular structures, entropy and trajectory-based constraints often produce a strict gap between abstract and Borel distinguishing numbers.

A plausible implication is that Borel combinatorics and entropy theory play a fundamental role in determining the achievable bounds for definable symmetry breaking in graphs arising from group actions and equivalence relations.

The paper highlights further open problems:

• Determining the exact Borel distinguishing number for shift graphs from more complicated groups (e.g., free group $\mathbb{F}_2$).
• Understanding how definability constraints interact with other distinguishing invariants (proper colorings, partitions, edge and total colorings).

6. Technical Formulations

Central definitions can be formalized as: $$D(G) = \min\left\{\,|I| : \exists c : X \to I\ \forall \varphi \neq \mathrm{id},\ \exists x,\ c(x) \neq c(\varphi(x))\,\right\},$$

$$D_B(G) = \min\left\{\,|I| : \exists\ \text{Borel}\ c : X\to I\ \forall \varphi \neq \mathrm{id},\ \exists x,\ c(x) \neq c(\varphi(x))\,\right\}.$$

Trajectory-based constraints used in shift graphs: $$\mathrm{tr}_f(\alpha) = (f(\sigma^i(\alpha)))_{i \in \mathbb{Z}},$$

$$\widehat{\delta}(i) = \delta(-i),\ \text{reflection condition.}$$

7. Connections to Classical Symmetry Breaking

While foundational combinatorial results (e.g., (Smith et al., 2011, Imrich et al., 2019)) establish that infinite locally finite graphs can have distinguishing number 2 via combinatorial constructions, these proofs typically do not yield Borel measurable colorings nor account for definable constraints. The techniques developed for the Borel context in (Bilge et al., 11 Aug 2025) require fundamentally different tools—often invoking descriptive set theory, Borel equivalence relations, and entropy calculations. The results sharply demarcate the gap between abstract and definably constrained symmetry breaking, and underscore the rich interplay between infinite graph combinatorics, group actions, and definability.

This comprehensive analysis reflects the current state of research in Borel distinguishing numbers, illuminating both their foundational importance and their new challenges in descriptive combinatorics.