Embedding tetrahedra into quasirandom hypergraphs

TL;DR

This paper proves that sufficiently large quasirandom hypergraphs with density greater than 1/2 necessarily contain a tetrahedron, advancing understanding of extremal configurations in hypergraph theory.

Contribution

It establishes the existence of tetrahedra in large quasirandom hypergraphs with density above 1/2, confirming a conjecture related to Erdős's question.

Findings

Large quasirandom hypergraphs with density > 1/2 contain tetrahedra.

The density threshold of 1/2 is optimal, as shown by known constructions.

The result connects quasirandomness with extremal hypergraph configurations.

Abstract

We investigate extremal problems for quasirandom hypergraphs. We say that a $3$-uniform hypergraph $H=(V,E)$ is $(d,\eta)$-quasirandom if for any subset $X\subseteq V$ and every set of pairs $P\subseteq V\times V$ the number of pairs $(x,(y,z))\in X\times P$ with $\{x,y,z\}$ being a hyperedge of $H$ is in the interval $d|X||P|\pm\eta|V|^3$. We show that for any $\varepsilon>0$ there exists $\eta>0$ such that every sufficiently large $(1/2+\varepsilon,\eta)$-quasirandom hypergraph contains a tetrahedron, i.e., four vertices spanning all four hyperedges. A known random construction shows that the density $1/2$ is best possible. This result is closely related to a question of Erd\H{o}s, whether every weakly quasirandom $3$-uniform hypergraph $H$ with density bigger than $1/2$, i.e., every large subset of vertices induces a hypergraph with density bigger than $1/2$, contains a tetrahedron.