A semi-algebraic version of Zarankiewicz's problem

TL;DR

This paper establishes tight bounds on the maximum edges in semi-algebraic bipartite graphs avoiding complete bipartite subgraphs, generalizing classical incidence theorems and employing advanced algebraic and combinatorial tools.

Contribution

It introduces tight bounds for edges in semi-algebraic bipartite graphs in low and high dimensions, extending the Szemerédi-Trotter theorem to a broader algebraic setting.

Findings

Maximum edges in $K_{k,k}$-free semi-algebraic graphs in $ ext{R}^2$ are $O((mn)^{2/3} + m + n)$.

In higher dimensions, the maximum edges are $C((mn)^{d/(d+1)+ ext{ε}} + m + n)$ for fixed $d$, $k$, and $ ext{ε}$.

Applications include point-variety incidence bounds and improved Erdős distance bounds.

Abstract

A bipartite graph $G$ is semi-algebraic in $\mathbb{R}^d$ if its vertices are represented by point sets $P,Q \subset \mathbb{R}^d$ and its edges are defined as pairs of points $(p,q) \in P\times Q$ that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in $2d$ coordinates. We show that for fixed $k$, the maximum number of edges in a $K_{k,k}$-free semi-algebraic bipartite graph $G = (P,Q,E)$ in $\mathbb{R}^2$ with $|P| = m$ and $|Q| = n$ is at most $O((mn)^{2/3} + m + n)$, and this bound is tight. In dimensions $d \geq 3$, we show that all such semi-algebraic graphs have at most $C\left((mn)^{ \frac{d}{d+1} + \varepsilon} + m + n\right)$ edges, where here $\varepsilon$ is an arbitrarily small constant and $C = C(d,k,t,\varepsilon)$. This result is a far-reaching generalization of the classical Szemer\'edi-Trotter incidence theorem. The proof combines tools from several fields: VC-dimension and shatter functions, polynomial partitioning, and Hilbert polynomials. We also present various applications of our theorem. For example, a general point-variety incidence bound in $\mathbb{R}^d$, an improved bound for a $d$-dimensional variant of the Erd\H{o}s unit distances problem, and more.