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Zarankiewicz bounds from distal regularity lemma

Mervyn Tong

TL;DR

This work broadens Zarankiewicz-type bounds from semialgebraic relations to all relations obeying the distal regularity lemma, a robust generalization of Szemerédi regularity applicable in distal structures. It introduces strong distal regularity tuples and proves a sharp binary-case bound, then extends to general $k$-ary relations, yielding a bound of the form $|E(P_1,\dots,P_k)| \ll_{u,\bar c,\lambda,\varepsilon} F^{\varepsilon}_{\bar c}(n_1,\dots,n_k)$ with appropriately defined $F^{\varepsilon}_{\bar c}$. The paper situates these results within distal model theory, showing that distal-regularity bounds apply to distal structures (including o-minimal and $p$-adic settings) and discussing limitations via an incidence-example that lacks a distal expansion. Overall, it unifies and extends prior semialgebraic Zarankiewicz bounds, providing a flexible framework for incidence bounds in a broad class of combinatorial geometries.

Abstract

Since Kővári, Sós, and Turán 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 Szemerédi regularity lemma satisfied by relations definable in distal structures (a vast generalisation of o-minimal structures).

Zarankiewicz bounds from distal regularity lemma

TL;DR

This work broadens Zarankiewicz-type bounds from semialgebraic relations to all relations obeying the distal regularity lemma, a robust generalization of Szemerédi regularity applicable in distal structures. It introduces strong distal regularity tuples and proves a sharp binary-case bound, then extends to general -ary relations, yielding a bound of the form with appropriately defined . The paper situates these results within distal model theory, showing that distal-regularity bounds apply to distal structures (including o-minimal and -adic settings) and discussing limitations via an incidence-example that lacks a distal expansion. Overall, it unifies and extends prior semialgebraic Zarankiewicz bounds, providing a flexible framework for incidence bounds in a broad class of combinatorial geometries.

Abstract

Since Kővári, Sós, and Turán 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 Szemerédi regularity lemma satisfied by relations definable in distal structures (a vast generalisation of o-minimal structures).

Paper Structure

This paper contains 14 sections, 12 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The semialgebraic case
  3. Structure of the paper
  4. Notation and basic definitions
  5. Indexing
  6. Relations
  7. Covers and partitions
  8. Asymptotics
  9. Acknowledgements
  10. Funding statement
  11. (Strong) distal regularity tuples
  12. Binary relations
  13. The general case
  14. Distality

Key Result

Theorem 1.1

Let $E(x, y)$ be a semialgebraic relation on $\mathbb{R}$ with description complexity at most $t$. Let $d_1:=|x|$ and $d_2:=|y|$. Then, for all finite $P\subseteq \mathbb{R}^x$ and $Q\subseteq \mathbb{R}^y$ with $m:=|P|$ and $n:=|Q|$, if $E(P,Q)$ is $K_{u,u}$-free, then for all $\varepsilon>0$ we ha

Figures (1)

  • Figure 1: The $\phi$-types over $B$

Theorems & Definitions (34)

  • Theorem 1.1: Fox--Pach--Sheffer--Suk--Zahl fox
  • Theorem 1.2: Do do
  • Definition 2.1
  • Theorem 2.2: Szemerédi, 1978 szemerediregularitylemma
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5: $\mathcal{M}=\mathbb{R}$: Fox--Pach--Suk, 2016 semialgebraicregularity; $\mathcal{M}$ distal: Chernikov--Starchenko, 2018 regularitylemma
  • Lemma 2.6
  • proof
  • ...and 24 more