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

TL;DR

This paper develops a Borel version of the Lovász Local Lemma applicable to graphs with finite asymptotic separation index, enabling Borel solutions for certain combinatorial problems in this setting.

Contribution

It introduces a Borel analogue of the Local Lemma for graphs with finite asymptotic separation index, bridging probabilistic combinatorics and Borel graph theory.

Findings

Borel version of the Lovász Local Lemma established

Borel solutions for locally checkable problems on graphs with finite asymptotic separation index

Extension of Brooks's theorem to this class of graphs

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 Lov\'asz 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.