Hypergraphs without complete partite subgraphs

TL;DR

This paper establishes new bounds on the Zarankiewicz number for certain complete r-partite r-uniform hypergraphs, improving previous results and introducing a novel approach using Behrend's construction.

Contribution

It provides improved bounds for Zarankiewicz numbers for hypergraphs with small parts, utilizing a new method based on sets with no 3-term arithmetic progression.

Findings

Zarankiewicz number for K(s_1,...,s_{r-1}, t) is n^{r-1/s - o(1)} for t > 3^{s+o(s)}

New bounds are achieved for small s_i, e.g., z(n, K(2,2,7))=n^{11/4 - o(1)}

Previous bounds required much larger t, e.g., ((r-1)(s-1))!

Abstract

Fix integers $r \ge 2$ and $1\le s_1\le \cdots \le s_{r-1}\le t$ and set $s=\prod_{i=1}^{r-1}s_i$. Let $K=K(s_1, \ldots, s_{r-1}, t)$ denote the complete $r$-partite $r$-uniform hypergraph with parts of size $s_1, \ldots, s_{r-1}, t$. We prove that the Zarankiewicz number $z(n, K)= n^{r-1/s-o(1)}$ provided $t> 3^{s+o(s)}$. Previously this was known only for $t > ((r-1)(s-1))!$ due to Pohoata and Zakharov. Our novel approach, which uses Behrend's construction of sets with no 3 term arithmetic progression, also applies for small values of $s_i$, for example, it gives $z(n, K(2,2,7))=n^{11/4-o(1)}$ where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.