Regular induced subgraphs of a random Graph

An old problem of Erdos, Fajtlowicz, and Staton asks for the order of a largest induced regular subgraph that can be found in every graph on $n$ vertices. Motivated by this problem, we consider the order of such a subgraph in a typical graph on $n$ vertices, i.e., in a binomial random graph $G(n,1/2)$. We prove that with high probability a largest induced regular subgraph of $G(n,1/2)$ has about $n^{2/3}$ vertices.