# Large cliques in hypergraphs with forbidden substructures

## TL;DR

This paper generalizes a classical graph theory result about large cliques in graphs without certain substructures to hypergraphs, with implications for discrete geometry and combinatorial theorems.

## Contribution

It introduces a higher-dimensional analogue of an induced subgraph concept, extending the result to k-uniform hypergraphs and linking combinatorics with geometric theorems.

## Key findings

- Generalized the Erdős–Gyárfás–Hubenko–Solymosi result to hypergraphs
- Established a combinatorial derivation of the fractional Helly theorem
- Connected the colorful Helly theorem with hypergraph properties

## Abstract

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 $c\binom{n}{2}$ 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.