OnK4-free subgraphs of random graphs

For 0<γ≤1 and graphsG andH, writeG→γH if any γ-proportion of the edges ofG spans at least one copy ofH inG. As customary, writeKr for the complete graph onr vertices. We show that for every fixed real η>0 there exists a constantC=C(η) such that almost every random graphGn,p withp=p(n)≥Cn−2/5 satisfiesGn,p→2/3+ηK4. The proof makes use of a variant of Szemerédi's regularity lemma for sparse graphs and is based on a certain superexponential estimate for the number of pseudo-random tripartite graphs whose triangles are not too well distributed. Related results and a general conjecture concerningH-free subgraphs of random graphs in the spirit of the Erdős-Stone theorem are discussed.