Dense subgraphs in random graphs

Abstract

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 γk2 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)=γlogγp+(1−γ)log1−γ1−p, we show that ωγ(Gn,p) is one of the two integers closest to 2α(γ,p)(logn−loglogn+logeα(γ,p)2)+12, 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.