  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.
