Zarankiewicz bounds from distal regularity lemma

TL;DR

This paper demonstrates that Zarankiewicz bounds, initially established for specific graph classes, extend to all relations satisfying the distal regularity lemma, broadening the scope of combinatorial bounds in model-theoretic structures.

Contribution

It generalizes Zarankiewicz bounds to relations satisfying the distal regularity lemma, expanding their applicability beyond semialgebraic relations.

Findings

Zarankiewicz bounds apply to all distal regularity lemma relations.

Extends bounds from semialgebraic to distal structures.

Provides a model-theoretic framework for combinatorial bounds.

Abstract

Since K\H{o}v\'ari, S\'os, and Tur\'an proved upper bounds for the Zarankiewicz problem in 1954, much work has been undertaken to improve these bounds, and some have done so by restricting to particular classes of graphs. In 2017, Fox, Pach, Sheffer, Suk, and Zahl proved better bounds for semialgebraic binary relations, and this work was extended by Do in the following year to arbitrary semialgebraic relations. In this paper, we show that Zarankiewicz bounds in the shape of Do's are enjoyed by all relations satisfying the distal regularity lemma, an improved version of the Szemer\'edi regularity lemma satisfied by relations definable in distal structures (a vast generalisation of o-minimal structures).