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

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 $\frac{1}{2}(k-1)$. This affirmatively answers a question of Erd\H{o}s and Pach from 1983.