On a generalisation of Mantel’s Theorem to Uniformly Dense Hypergraphs

For a |k|-uniform hypergraph |F| let |$ex(n,F)$| be the maximum number of edges of a |k|-uniform |n|-vertex hypergraph |H| which contains no copy of |F|⁠. Determining or estimating |$ex(n,F)$| 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 Erdős and Sós about 30 years ago. We show that every |k|-uniform hypergraph |H| with density |>2¹⁻ᵏ| with respect to every large collection of |k|-cliques induced by sets of |$(k-2)$|-tuples contains a copy of |F|⁠. The required density |2¹⁻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≥ 2| utilises the regularity method for hypergraphs.