Bipartite Ramsey numbers involving large Kn,nKn,n

Let br(H1,H2)br(H1,H2) be the bipartite Ramsey number for bipartite graphs H1H1 and H2H2. It is shown that the order of magnitude of br(Kt,n,Kn,n)br(Kt,n,Kn,n) is nt+1/(logn)tnt+1/(logn)t for t≥1t≥1 fixed and n→∞n→∞. Moreover, if HH is an isolate-free bipartite graph of order hh having bipartition (A,B)(A,B) that satisfies Δ(B)≤tΔ(B)≤t, then br(H,Kn,n)br(H,Kn,n) can be bounded from above by (hn/logn)t(logn)α(t)(hn/logn)t(logn)α(t) for large nn, where α(1)=α(2)=1α(1)=α(2)=1 and α(t)=0α(t)=0 for t≥3t≥3.