Asymptotic Size Ramsey Results for Bipartite Graphs

We show that limn→∞ r̂(F1,n, . . . , Fq,n, Fp+1, . . . , Fr)/n exists, where the bipartite graphs Fq+1, . . . , Fr do not depend on n while, for 1 ≤ i ≤ q, Fi,n is obtained from some bipartite graph Fi with parts V1 ∪ V2 = V (Fi) by duplicating each vertex v ∈ V2 (cv + o(1))n times for some real cv > 0. In fact, the limit is the minimum of a certain mixed integer program. Using the Farkas Lemma we compute it when each forbidden graph is a complete bipartite graph, in particular answering a question of Erdős, Faudree, Rousseau and Schelp (1978) who asked for the asymptotics of r̂(Ks,n, Ks,n) for fixed s and large n. Furthermore, we prove (for all sufficiently large n) the conjecture of Faudree, Rousseau and Sheehan (1983) that r̂(K2,n, K2,n) = 18n − 15.