Tur\'an density of cliques of order five in $3$-uniform hypergraphs with quasirandom links

TL;DR

This paper proves that 3-uniform hypergraphs with quasirandom link graphs exceeding a density of 1/3 necessarily contain a 5-vertex clique, establishing an asymptotically optimal threshold.

Contribution

It establishes the exact asymptotic threshold for the existence of 5-vertex cliques in quasirandom 3-uniform hypergraphs based on link graph density.

Findings

Hypergraphs with link density > 1/3 contain 5-cliques

The threshold of 1/3 is asymptotically optimal

Provides a characterization of clique existence in quasirandom hypergraphs

Abstract

We show that $3$-uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than $1/3$ contain a clique on five vertices. This result is asymptotically best possible.