Consider the triangle-free process, which is defined as follows. Start with G(0), an empty graph on n vertices. Given G(i - 1), let G(i) = G(i - 1) boolean OR {g(i)}, where g(i) is an edge that is chosen uniformly at random from the set of edges that are not in G(i - 1) and can be added to G(i - 1) without creating a triangle. The process ends once a maximal triangle-free graph has been created. Let H be a fixed triangle-free graph and let X(H) (i) count the number of copies of H in G(i). We give an asymptotically sharp estimate for E(X(H) (i)), for every 1 << i <= 2(-5)n(3/2)root ln n, at the limit as n -> infinity. Moreover, we provide conditions that guarantee that a.a.s. X(H) (i) = 0, and that X(H) (i) is concentrated around its mean. (C) 2011 Wiley Periodicals, Inc. Random Struct. Alg., 39, 539-543, 2011
