On a generalisation of Mantel's theorem to uniformly dense hypergraphs

Abstract

For a $k$-uniform hypergraph $F$ let $\text{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\text{ex}(n,F)$ is a classical and central problem in extremal combinatorics. While for $k=2$ this problem is well understood, due to the work of Turán and of Erdős and Stone, only very little is known for $k$-uniform hypergraphs for $k>2$. We focus on the case when $F$ is a $k$-uniform hypergraph with three edges on $k+1$ vertices. Already this very innocent (and maybe somewhat particular looking) problem is still wide open even for $k=3$. We consider a variant of the problem where the large hypergraph $H$ enjoys additional hereditary density conditions. Questions of this type were suggested by Erd\H os and Sós about 30 years ago. We show that every $k$-uniform hypergraph $H$ with density $>2^{1-k}$ with respect to every large collections of $k$-cliques induced by sets of $(k-2)$-tuples contains a copy of $F$. The required density $2^{1-k}$ is best possible as higher order tournament constructions show. Our result can be viewed as a common generalisation of the first extremal result in graph theory due to Mantel (when $k=2$ and the hereditary density condition reduces to a normal density condition) and a recent result of Glebov, Král', and Volec (when $k=3$ and large subsets of vertices of $H$ induce a subhypergraph of density $>1/4$). Our proof for arbitrary $k\ge 2$ utilises the regularity method for hypergraphs.

38 pages, second version addresses changes arising from the referee reports

References in corpus (2)

• On a Turán problem in weakly quasirandom 3-uniform hypergraphs
• Embedding tetrahedra into quasirandom hypergraphs

Cited by in corpus (2)

• Extremal problems in uniformly dense hypergraphs
• Hypergraphs with uniform Turán density equal to 8/27