The Triangle-Free Process

Consider the following stochastic graph process. We begin with G0, the empty graph on n vertices, and form Gi by adding a randomly chosen edge ei to Gi−1 where ei is chosen uniformly at random from the collection of pairs of vertices that neither appear as edges in Gi−1 nor form triangles when added as edges to Gi−1. Let the random variable M be the number of edges in the maximal triangle free graph generated by this process. We prove that asymptotically almost surely M = Θ(n √ logn). This resolves a conjecture of Spencer. Furthermore, the independence number of GM is asymptotically almost surely Θ( √ n logn), which implies that the Ramsey number R(3, t) is bounded below by a constant times t/ log t (a fact that was previously established by Jeong Han Kim). The methods introduced here extend to the K4-free process, thereby establishing the bound R(4, t) = Ω(t/ log t).