Hyperfiniteness and Borel combinatorics

TL;DR

This paper explores the interplay between hyperfiniteness and Borel graph combinatorics, providing new results on chromatic numbers, perfect matchings, and measure properties, using game-theoretic techniques and addressing longstanding questions.

Contribution

It introduces novel methods to analyze hyperfinite Borel graphs, computes chromatic numbers, constructs specific examples, and connects these to measure and ultrafilter problems, advancing understanding in descriptive combinatorics.

Findings

Computed Borel chromatic and edge chromatic numbers for hyperfinite graphs.

Constructed hyperfinite graphs with no Borel perfect matching.

Demonstrated failures of the Borel local lemma in certain hyperfinite graphs.

Abstract

We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel chromatic number and measure chromatic number. We also show that for every $d > 1$ there is a $d$-regular acyclic hyperfinite Borel bipartite graph with no Borel perfect matching. These techniques also give examples of hyperfinite bounded degree Borel graphs for which the Borel local lemma fails, in contrast to the recent results of Cs\'oka, Grabowski, M\'ath\'e, Pikhurko, and Tyros. Related to the Borel Ruziewicz problem, we show there is a continuous paradoxical action of $(\mathbb{Z}/2\mathbb{Z})^{*3}$ on a Polish space that admits a finitely additive invariant Borel probability measure, but admits no countably additive invariant Borel probability measure. In the context of studying ultrafilters on the quotient space of equivalence relations under $\mathrm{AD}$, we also construct an ultrafilter $U$ on the quotient of $E_0$ which has surprising complexity. In particular, Martin's measure is Rudin-Kiesler reducible to $U$. We end with a problem about whether every hyperfinite bounded degree Borel graph has a witness to its hyperfiniteness which is uniformly bounded below in size.