Extremal problems in uniformly dense hypergraphs

Abstract

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem in extremal combinatorics. While for graphs ($k=2$) this problem is well understood, due to the work of Mantel, Tur\'an, Erd\H{o}s, Stone, Simonovits and many others, only very little is known for $k$-uniform hypergraphs for $k>2$. Already the case when $F$ is a $k$-uniform hypergraph with three edges on $k+1$ vertices is still wide open even for $k=3$. We consider variants of such problems where the large hypergraph $H$ enjoys additional hereditary density conditions. Questions of this type were suggested by Erd\H{o}s and S\'os about 30 years ago. In recent work with R\"odl and Schacht it turned out that the regularity method for hypergraphs, established by Gowers and by R\"odl et al. about a decade ago, is a suitable tool for extremal problems of this type and we shall discuss some of those recent results and some interesting open problems in this area.