Cutting lemma and Zarankiewicz's problem in distal structures

TL;DR

This paper develops a cutting lemma for definable sets in distal structures and extends Zarankiewicz's problem and the Szemerédi-Trotter theorem to o-minimal structures, broadening combinatorial geometry in model theory.

Contribution

It introduces a cutting lemma for distal structures and generalizes planar Zarankiewicz and Szemerédi-Trotter results to o-minimal frameworks.

Findings

Established a cutting lemma for definable families in distal structures

Proved the optimality of distal cell decomposition on the plane

Extended Zarankiewicz's problem and Szemerédi-Trotter theorem to o-minimal structures

Abstract

We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in $o$-minimal expansions of fields. Using it, we generalize the results in [J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl. "A semi-algebraic version of Zarankiewicz's problem"] on the semialgebraic planar Zarankiewicz problem to arbitrary $o$-minimal structures, in particular obtaining an $o$-minimal generalization of the Szemer\'edi-Trotter theorem.