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 XH(i) count the number of copies of H in G(i). We give an asymptotically sharp estimate for E(XH(i)), for every 1 i ≤ 2−5n3/2 √ lnn, at the limit as n→∞. Moreover, we provide conditions that guarantee that a.a.s. XH(i) = 0, and that XH(i) is concentrated around its mean.