The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as $n \to \infty$, the minimum degree in G is at least $cn^{1-(v_H-2)/(e_H-1)}(\log n)^{1/(e_H-1)}$. This gives new lower bounds for the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite graphs $K_{r,r}$ with $r \ge 5$. When H is a complete graph $K_s$ with $s \ge 5$ we show that for some C>0, with high probability the independence number of G is at most $Cn^{2/(s+1)}(\log n)^{1-1/(e_H-1)}$. This gives new lower bounds for Ramsey numbers R(s,t) for fixed $s \ge 5$ and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.Comment: 36 page

Consider the following stochastic graph process. We begin with G 0 , the empty graph on n vertices, and form G i by adding a randomly chosen edge e i to G i−1 where e i is chosen uniformly at random from the collection of pairs of vertices that neither appear as edges in G i−1 nor form triangles when added as edges to G i−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 3/2 √ log n). This resolves a conjecture of Spencer. Furthermore, the independence number of G M is asymptotically almost surely Θ( √ n log n), which implies that the Ramsey number R(3, t) is bounded below by a constant times t 2 / log t (a fact that was previously established by Jeong Han Kim). The methods introduced here extend to the K 4 -free process, thereby establishing the bound R(4, t) = Ω(t 5/2 / log 2 t).

This paper investigates the number of random edges required to add to an arbitrary dense graph in order to make the resulting graph hamiltonian with high probability. Adding Θ(n) random edges is both necessary and sufficient to ensure this for all such dense graphs. If, however, the original graph contains no large independent set, then many fewer random edges are required. We prove a similar result for directed graphs.

The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal triangle-free graph at which the triangle-free process terminates. We also bound the independence number of this graph, which gives an improved lower bound on the Ramsey numbers R(3, t): we show R(3, t) > (1/4 − o(1))t 2 / log t, which is within a 4 + o(1) factor of the best known upper bound. Our improvement on previous analyses of this process exploits the self-correcting nature of key statistics of the process. Furthermore, we determine which bounded size subgraphs are likely to appear in the maximal triangle-free graph produced by the triangle-free process: they are precisely those triangle-free graphs with 2-density at most 2.

Let e1, e′1; e2, e′2;…;ei, e′i;⋅⋅⋅ be a sequence of ordered pairs of edges chosen uniformly at random from the edge set of the complete graph Kn (i.e. we sample with replacement). This sequence is used to form a graph by choosing at stage i, i=1,…, one edge from ei,e′i to be an edge in the graph, where the choice at stage i is based only on the observation of the edges that have appeared by stage i. We show that these choices can be made so that whp the size of the largest component of the graph formed at stage 0.535n is polylogarithmic in n. This resolves a question of Achlioptas. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 75–85, 2001