On K-free Subgraphs of Random Graphs
نویسندگان
چکیده
For 0 < γ ≤ 1 and graphs G and H, write G→γ H if any γ-proportion of the edges of G span at least one copy of H in G. As customary, write Kr for the complete graph on r vertices. We show that for every fixed real η > 0 there exists a constant C = C(η) such that almost every random graph Gn,p with p = p(n) ≥ Cn−2/5 satisfies Gn,p →2/3+η K4. 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 concerning H-free subgraphs of random graphs in the spirit of the Erdős–Stone–Simonovits theorem are discussed.
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