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Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index

Anton Bernshteyn, Felix Weilacher

TL;DR

This work develops Borel analogues of the Lovász Local Lemma under finite asymptotic separation index and ties them to efficient randomized LOCAL algorithms. By introducing a shattering-based LLL and leveraging structured-graph reductions, it yields constructive Borel colorings and edge-colorings for broad classes of Borel graphs with finite asi, including Schreier graphs. Key contributions include Borel Brooks/Johansson-type bounds, near-Delta edge-colorings, and Schreier-graph Vizing-type results, all grounded in an LLL/LOCAL pipeline that connects distributed computing to descriptive combinatorics. The results substantially broaden when and where Borel combinatorics can mirror finite/classical behavior, with implications for measurable and Baire-measurable contexts and potential for further generalizations in structured graphs and hyperfinite settings.

Abstract

Asymptotic separation index is a parameter that measures how easily a Borel graph can be approximated by its subgraphs with finite components. In contrast to the more classical notion of hyperfiniteness, asymptotic separation index is well-suited for combinatorial applications in the Borel setting. The main result of this paper is a Borel version of the Lovász Local Lemma -- a powerful general-purpose tool in probabilistic combinatorics -- under a finite asymptotic separation index assumption. As a consequence, we show that locally checkable labeling problems that are solvable by efficient randomized distributed algorithms admit Borel solutions on bounded degree Borel graphs with finite asymptotic separation index. From this we derive a number of corollaries, for example a Borel version of Brooks's theorem for graphs with finite asymptotic separation index.

Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index

TL;DR

This work develops Borel analogues of the Lovász Local Lemma under finite asymptotic separation index and ties them to efficient randomized LOCAL algorithms. By introducing a shattering-based LLL and leveraging structured-graph reductions, it yields constructive Borel colorings and edge-colorings for broad classes of Borel graphs with finite asi, including Schreier graphs. Key contributions include Borel Brooks/Johansson-type bounds, near-Delta edge-colorings, and Schreier-graph Vizing-type results, all grounded in an LLL/LOCAL pipeline that connects distributed computing to descriptive combinatorics. The results substantially broaden when and where Borel combinatorics can mirror finite/classical behavior, with implications for measurable and Baire-measurable contexts and potential for further generalizations in structured graphs and hyperfinite settings.

Abstract

Asymptotic separation index is a parameter that measures how easily a Borel graph can be approximated by its subgraphs with finite components. In contrast to the more classical notion of hyperfiniteness, asymptotic separation index is well-suited for combinatorial applications in the Borel setting. The main result of this paper is a Borel version of the Lovász Local Lemma -- a powerful general-purpose tool in probabilistic combinatorics -- under a finite asymptotic separation index assumption. As a consequence, we show that locally checkable labeling problems that are solvable by efficient randomized distributed algorithms admit Borel solutions on bounded degree Borel graphs with finite asymptotic separation index. From this we derive a number of corollaries, for example a Borel version of Brooks's theorem for graphs with finite asymptotic separation index.
Paper Structure (21 sections, 49 theorems, 91 equations)

This paper contains 21 sections, 49 theorems, 91 equations.

Table of Contents

  1. Introduction and main results
  2. Borel combinatorics and the asymptotic separation index
  3. The Lovász Local Lemma and its Borel versions
  4. Distributed algorithms
  5. Combinatorial applications
  6. Bounds on the Borel chromatic number
  7. Borel edge-colorings
  8. Borel edge-colorings of Schreier graphs
  9. Borel independent complete sections
  10. Preliminaries
  11. The spaces of finite sets and constraints
  12. Borel combinatorics on component-finite graphs
  13. Proofs of the main results
  14. Proof of Theorem \ref{['theo:main_easy_asi']}
  15. From the LLL to $\mathsf{LOCAL}$ algorithms
  16. ...and 6 more sections

Key Result

Theorem 1.5

If $G$ is a Borel graph of finite maximum degree $\Delta$, then $\chi_\mathsf{B}(G) \leqslant \Delta + 1$.

Theorems & Definitions (87)

  • Definition 1.1: Colorings and the chromatic number
  • Definition 1.2: Borel graphs
  • Definition 1.4: Borel, measurable, Baire-measurable chromatic numbers
  • Theorem 1.5: Kechris--Solecki--Todorcevic KST
  • Theorem 1.6: Marks Marks
  • Theorem 1.7: Brooks Brooks[Theorem 5.2.4]Diestel
  • Theorem 1.8: Johansson Joh_triangle[§13]MolloyReed
  • Definition 1.9: Hyperfinite graphs
  • Theorem 1.10: Conley--Jackson--Marks--Seward--Tucker-Drob HypComb
  • Definition 1.11: Asymptotic separation index
  • ...and 77 more