A note on the Erd\H{o}s-Hajnal property for stable graphs

TL;DR

This paper offers a simplified proof of the Erdős-Hajnal conjecture for stable graphs and hypergraphs lacking the k-order property, building on prior complex proofs by Malliaris and Shelah.

Contribution

It provides a more accessible proof of the Erdős-Hajnal conjecture for stable graphs and hypergraphs without the k-order property, simplifying previous complex arguments.

Findings

Proof of the Erdős-Hajnal conjecture for stable graphs

Extension to hypergraphs without the k-order property

Simplification of existing proof techniques

Abstract

In this short note we provide a relatively simple proof of the Erd\H{o}s-Hajnal conjecture for families of finite (hyper-)graphs without the $k$-order property. It was originally proved by M. Malliaris and S. Shelah in "Regularity lemmas for stable graphs", Transactions AMS, 366, 2014, 1551-1585.