Zarankiewicz's problem for semi-algebraic hypergraphs

TL;DR

This paper extends Zarankiewicz's problem to semi-algebraic hypergraphs, providing upper bounds on the number of hyperedges without certain complete subhypergraphs, with applications to geometric problems.

Contribution

It introduces upper bounds for hyperedges in semi-algebraic hypergraphs avoiding specific complete subhypergraphs, generalizing previous results from graphs to hypergraphs.

Findings

Established bounds for k=2 matching previous work.

Derived new bounds for k=3 hypergraphs involving parts sizes and dimension.

Applied bounds to geometric problems like the unit area and minor problems.

Abstract

Zarankiewicz's problem asks for the largest possible number of edges in a graph that does not contain a $K_{u,u}$ subgraph for a fixed positive integer $u$. Recently, Fox, Pach, Sheffer, Sulk and Zahl considered this problem for semi-algebraic graphs, where vertices are points in $\mathbb{R}^d$ and edges are defined by some semi-algebraic relations. In this paper, we extend this idea to semi-algebraic hypergraphs. For each $k\geq 2$, we find an upper bound on the number of hyperedges in a $k$-uniform $k$-partite semi-algebraic hypergraph without $K_{u_1,\dots,u_k}$ for fixed positive integers $u_1,\dots, u_k$. When $k=2$, this bound matches the one of Fox et.al. and when $k=3$, it is $$O\left((mnp)^{\frac{2d}{2d+1}+\varepsilon}+m(np)^{\frac{d}{d+1}+\varepsilon}+n(mp)^{\frac{d}{d+1}+\varepsilon}+p(mn)^{\frac{d}{d+1}+\varepsilon}+mn+np+pm\right),$$ where $m,n,p$ are the sizes of the parts of the tripartite hypergraph and $\varepsilon$ is an arbitrarily small positive constant. We then present applications of this result to a variant of the unit area problem, the unit minor problem and intersection hypergraphs.