Large Cliques in Hypergraphs with Forbidden Substructures

Abstract

A result due to Gyarfas, Hubenko, and Solymosi (answering a question of Erdős) states that if a graph G on n vertices does not contain K2,2 as an induced subgraph yet has at least $$c\left(\begin{array}{c}n 2\end{array}\right)$$ 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 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.