Two Erdos-Hajnal-type theorems for forbidden order-size pairs

TL;DR

This paper establishes two Erdős-Hajnal-type theorems for hypergraphs, showing that the absence of certain substructures guarantees large homogeneous sets or many specific edge configurations.

Contribution

The paper introduces two variants of the Erdős-Hajnal conjecture for hypergraphs, proving the existence of large homogeneous sets and many specific edge configurations under certain conditions.

Findings

Hypergraphs without fixed edge counts have larger homogeneous sets.

3-graphs lacking polynomial-sized homogeneous sets contain many specific edge configurations.

Progress on a problem posed by Axenovich et al.

Abstract

The celebrated Erd\H{o}s-Hajnal conjecture says that any graph without a fixed induced subgraph $H$ contains a very large homogeneous set. A direct analog of this conjecture is not true for hypergraphs. In this paper we present two natural variants of this problem which do hold for hypergraphs. We show that for every $r \geq 3$, $m \geq m_0(r)$ and $0 \leq f \leq \binom{m}{r}$, if an $r$-graph $G$ does not contain $m$ vertices spanning exactly $f$ edges, then $G$ contains much bigger homogeneous sets than what is guaranteed to exist in general $r$-graphs. We also prove that if a $3$-graph $G$ does not contain homogeneous sets of polynomial size, then for every $m \geq 3$ there are $\Omega(m^3)$ values of $f$ such that $G$ contains $m$ vertices spanning exactly $f$ edges. This makes progress on a problem of Axenovich, Brada\v{c}, Gishboliner, Mubayi and Weber.