Triangle‐free subgraphs in the triangle‐free process

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) ∪{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 ??(X_H(i)), for every \begin{align*}1 \ll i \le 2^{-5} n^{3/2} \sqrt{\ln n}\end{align*}, at the limit as n →∞. Moreover, we provide conditions that guarantee that a.a.s. X_H(i) = 0, and that X_H(i) is concentrated around its mean.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011