Borel Combinatorics of Locally Finite Graphs

TL;DR

This paper introduces Borel combinatorics on topological spaces, exploring definable graphs and their properties, with new results on Borel coloring bounds and connections to other mathematical areas.

Contribution

It provides foundational tools and results in Borel combinatorics, including Borel versions of algorithms and the construction of graphs with optimal coloring bounds.

Findings

Borel versions of greedy algorithms and augmenting procedures

Construction of acyclic Borel graphs with optimal Borel chromatic number bound

Connections to LOCAL algorithms and measurable combinatorial theorems

Abstract

We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Borel versions of greedy algorithms and augmenting procedures, local rules, Borel transversals, etc. Also, we present the construction of Andrew Marks of acyclic Borel graphs for which the greedy bound $\Delta+1$ on the Borel chromatic number is best possible. In the remainder of the paper we briefly discuss various topics such as relations to LOCAL algorithms, measurable versions of Hall's marriage theorem and of Lov\'asz Local Lemma, applications to equidecomposability, etc.