OnK 4-free subgraphs of random graphs

Abstract

For 0 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 Szemeredi'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.