Multicolor Ramsey Numbers For Complete Bipartite Versus Complete Graphs

Let H1, . . . ,Hk be graphs. The multicolor Ramsey number r(H1, . . . ,Hk) is the minimum integer r such that in every edge-coloring of Kr by k colors, there is a monochromatic copy of Hi in color i for some 1 ≤ i ≤ k. In this paper, we investigate the multicolor Ramsey number r(K2,t, . . . ,K2,t,Km), determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of t and m. Several different constructions are used for the lower bounds, including the random graph and explicit graphs built from finite fields. A technique of Alon and Rödl using the probabilistic method and spectral arguments is employed to supply tight lower bounds. A sample result is c1 m2t log(mt) ≤ r(K2,t,K2,t,Km) ≤ c2 m2t log m for any t and m, where c1 and c2 are absolute constants.