For a constant γ∈[0,1] and a graph G, let ωγ(G) be
the largest integer k for which there exists a k-vertex subgraph of G
with at least γ(2k) edges. We show that if 0<p<γ<1 then
ωγ(Gn,p) is concentrated on a set of two integers. More
precisely, with
α(γ,p)=γlogpγ+(1−γ)log1−p1−γ,
we show that ωγ(Gn,p) is one of the two integers closest to
α(γ,p)2(logn−loglogn+log2eα(γ,p))+21, with high probability.
While this situation parallels that of cliques in random graphs, a new
technique is required to handle the more complicated ways in which these
"quasi-cliques" may overlap