On a Ramsey-type problem of Erd\H{o}s and Pach

TL;DR

This paper proves that for sufficiently large graphs, either the graph or its complement contains a large induced subgraph with high minimum degree, answering a longstanding question by Erdős and Pach from 1983.

Contribution

It establishes the existence of a universal constant C such that graphs of size Ck log k always contain a large high-minimum-degree induced subgraph or its complement.

Findings

Existence of constant C for large graphs

Either G or its complement has a high-degree induced subgraph

Answers Erdős and Pach's 1983 question

Abstract

In this paper we show that there exists a constant $C>0$ such that for any graph $G$ on $Ck\ln k$ vertices either $G$ or its complement $\bar{G}$ has an induced subgraph on $k$ vertices with minimum degree at least $\frac12(k-1)$. This affirmatively answers a question of Erd\H{o}s and Pach from 1983.