Sharp Bounds For Some Multicolor Ramsey Numbers

Let H ₁,H ₂, . . .,H _k+1 be a sequence of k+1 finite, undirected, simple graphs. The (multicolored) Ramsey number r(H ₁,H ₂,...,H _k+1) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k+1 colors, there is a monochromatic copy of H _i in color i for some 1≤i≤k+1. We describe a general technique that supplies tight lower bounds for several numbers r(H ₁,H ₂,...,H _k+1) when k≥2, and the last graph H _k+1 is the complete graph K _m on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K ₃,K ₃,K _m ) = Θ(m ³ poly logm), thus solving (in a strong form) a conjecture of Erdőos and Sós raised in 1979. Another special case of our result implies that r(C ₄,C ₄,K _m ) = Θ(m ² poly logm) and that r(C ₄,C ₄,C ₄,K _m ) = Θ(m ²/log² m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain estimates of character sums.