Large cliques in hypergraphs with forbidden substructures

A result due to Gy'arf'as, Hubenko, and Solymosi (answering a question of Erd"os) states that if a graph $G$ on $n$ vertices does not contain $K_{2,2}$ as an induced subgraph yet has at least $c\left(\genfrac{}{}{0px}{}{n}{2}\right)$ edges, then $G$ has a complete subgraph on at least $\frac{c^2}{10}n$ vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced $K_{2,2}$ which allows us to generalize their result to $k$-uniform hypergraphs. Our result also has an interesting consequence in discrete geometry. In particular, it implies that the fractional Helly theorem can be derived as a purely combinatorial consequence of the colorful Helly theorem.