Large Cliques in Hypergraphs with Forbidden Substructures

Abstract

A result due to Gyarfas, Hubenko, and Solymosi (answering a question of Erd}os) states that if a graph G on n vertices does not contain K2;2 as an induced subgraph yet has at least c n 2 edges, then G has a complete subgraph on at least c2 10 n vertices. In this paper we suggest a \higher-dimensional" analogue of the notion of an induced K2;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