Multicolor Ramsey Numbers for Complete Bipartite Versus Complete Graphs

Details Multicolor Ramsey Numbers for Complete Bipartite Versus Complete Graphs Multicolor Ramsey Numbers for Complete Bipartite Versus Complete Graphs

2012-01-10

arxiv.org/abs/1201.2123v1

Mathematics

Combinatorics

26 pages, 0 figures

Scientific paper

Let H_1, ..., H_k be graphs. The multicolor Ramsey number r(H_1,...,H_k) is the minimum integer r such that in every edge-coloring of K_r by k colors, there is a monochromatic copy of H_i in color i for some 1 <= i <= k. In this paper, we investigate the multicolor Ramsey number $r(K_{2,t},...,K_{2,t},K_m)$, 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 R\"odl using the probabilistic method and spectral arguments is employed to supply tight lower bounds. A sample result is $c_1 m^2t/\log^4(mt) \leq r(K_{2,t},K_{2,t},K_m) \leq c_2 m^2t/\log^2 m$ for some constants c_1 and c_2.

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