The Turán Theorem for Random Graphs

The aim of this paper is to prove a Turán type theorem for random graphs. For 0 < γ ≤ 1 and graphs G and H, write G→γ H if any γ-proportion of the edges of G spans at least one copy of H in G. We show that for every l ≥ 2 and every fixed real 1/(l − 1) > δ > 0 almost every graph G in the binomial random graph model G(n, q), with q = q(n) ((logn)4/n)1/(l−1), satisfies G→(l−2)/(l−1)+δ Kl, where Kl is the complete graph on l vertices. Our result naturally extends to the case where H is a d-degenerate graph. In this case we show that almost every graph G in G(n, q) with q = q(n) ((logn)4/n)1/d satisfies G →(χ(H)−2)/(χ(H)−1)+δ H, where as usual χ(H) denotes the chromatic number of H.