Complexity of Finite Borel Asymptotic Dimension

TL;DR

This work establishes that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\\mathbf{\\Sigma}^1_2$-complete and develops a combinatorial characterization for graphs generated by a single Borel function via forward-independent hitting sets. By translating geometric questions about asdim into purely combinatorial problems, the authors obtain a CSP dichotomy for digraphs generated in this way, showing $\\mathbf{\\Pi}^1_1$ or $\\mathbf{\\Sigma}^1_2$-complete outcomes depending on loops and ergodicity, and relate these to the classical CSP dichotomy. They also prove the equivalence of the coverings and equivalence-relations formulations of finite Borel asymptotic dimension in this setting and provide constructive links between hitting sets and asdim, culminating in a main theorem that ties the descriptive-set-theoretic complexity to forward-independent hitting sets. The results illuminate the interaction between measurable combinatorics, CSP theory, and distributed computing models, and raise natural extensions to bounded-degree graphs and hypersmoothness questions. Overall, the paper advances the understanding of the logical complexity of Borel combinatorial properties and their algorithmic interpretations in CSPs and the LOCAL model.

Abstract

We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\mathbfΣ^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated by a single Borel function. As an application of this characterization, we classify the complexities of digraph homomorphism problems for this class of graphs.

Figures (2)

• Figure 1: (a) A directed graph $H$ such that $\operatorname{CSP}^{\operatorname{function}}_B(H)$ is $\mathbf{\Pi}^1_1$. (b) An abstract walk $p$. (c) The directed graph $H^p$, which has the property that $\operatorname{CSP}^{\operatorname{function}}_B(H^p)$ is $\mathbf{\Sigma}^1_2$-complete, showing that $\operatorname{CSP}_B(H)$ is $\mathbf{\Sigma}^1_2$-complete as well
• Figure 2: The structure $\mathcal{D}_r$ on $\mathbb N$ for various values of $r \in \mathbb N^+$

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5: Connections to CSPs
• Remark 1.6: $\operatorname{CSP}_B(H)$ vs $\operatorname{CSP}^{\operatorname{function}}_B(H)$
• Remark 1.7: Connections to the LOCAL model
• Definition 2.1: ${\bf \Sigma}^1_2$-completeness
• Theorem 2.2: Theorem 3.1, fsv2024
• Theorem 2.3