Bipartite Ramsey Numbers and Zarankiewicz Numbers

The bipartite Ramsey number b(m, n) is the minimum b such that any 2-coloring of Kb,b results in a monochromatic Km,m subgraph in the first color or a monochromatic Kn,n subgraph in the second color. The Zarankiewicz number z(m, n; s, t) is the maximum size among Ks,t-free subgraphs of Km,n. In this work, we discuss the intimate relationship between the two numbers as well as propose a new method for bounding the Zarankiewicz numbers. We use the better bounds to improve the upper bound on b(2, 5), in addition we improve the lower bound of b(2, 5) by construction. The new bounds are shown to be 17 ≤ b(2, 5) ≤ 18. Additionally, we apply the same methods to the multicolor case b(2, 2, 3) which has previously not been studied and determine bounds to be 16 ≤ b(2, 2, 3) ≤ 23

Bipartite Ramsey numbers involving large Kn,n

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