Lower Bounds for the Size of Random Maximal H-Free Graphs

We consider the next random process for generating a maximal H-free graph: Given a fixed graph H and an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph Kn. Then, traverse the edges of Kn according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving n-vertex graph unless its addition creates a copy of H. The result of this process is a maximal H-free graph Mn(H). Our main result is a new lower bound on the expected number of edges in Mn(H), for H that is regular, strictly 2-balanced. As a corollary, we obtain new lower bounds for Turán numbers of complete, balanced bipartite graphs. Namely, for fixed r ≥ 5, we show that ex(n,Kr,r) = Ω(n 2−2/(r+1)(ln lnn)1/(r 2−1)). This improves an old lower bound of Erdős and Spencer. Our result relies on giving a non-trivial lower bound on the probability that a given edge is included in Mn(H), conditioned on the event that the edge is traversed relatively (but not trivially) early during the process.