Bipartite Ramsey numbers involving large Kn,n

Let br(H-1, H-2) be the bipartite Ramsey number for bipartite graphs H-1 and H-2. It is shown that the order of magnitude of br(K-t,K-n, K-n,K-n) is n(t+1)/(log n)(t) for t >= 1 fixed and n -> infinity. Moreover, if H is an isolate-free bipartite graph of order h having bipartition (A, B) that satisfies Delta(B) <= t. then br(H, K-n,K-n) can be bounded from above by (hn/log n)(t) (log n)(alpha(t)) for large n, where alpha(1) = alpha(2) = 1 and alpha(t) = 0 for t >= 3. (C) 2008 Elsevier Ltd. All rights reserved.