Abstract
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, writeK r 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 graphG n,p withp=p(n)≥Cn −2/5 satisfiesG n,p →2/3+η K 4. 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.
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The first author was partially supported by FAPESP (Proc. 93/0603-1) and by CNPq (Proc. 300334/93-1 and ProTeM-CC-II Project ProComb). Part of this work was done while the second author was visiting the University of São Paulo, supported by FAPESP (Proc. 94/4276-8). The third author was partially supported by the NSF grant DMS-9401559.