Multicolor Ramsey Numbers for Complete Bipartite Versus Complete Graphs

TL;DR

This paper determines the asymptotic behavior of multicolor Ramsey numbers for complete bipartite versus complete graphs, using probabilistic, algebraic, and spectral methods to establish bounds.

Contribution

It provides new asymptotic bounds for multicolor Ramsey numbers involving bipartite and complete graphs, employing novel constructions and techniques.

Findings

Established asymptotic bounds for r(K_{2,t},...,K_{2,t},K_m)

Used probabilistic and algebraic constructions for lower bounds

Applied spectral methods for tight bounds

Abstract

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 any t and m, where c_1 and c_2 are absolute constants.