On a Ramsey-type problem of Erdős and Pach

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 Erdős and Pach from 1983.

9 pages; in 2nd version, Eoin Long has been added as co-author and the main result is improved by a log k factor