Asymptotically tight bounds for some multicolored Ramsey numbers

Let H1,H2, . . . ,Hk+1 be a sequence of k + 1 finite, undirected, simple graphs. The (multicolored) Ramsey number r(H1,H2, . . . ,Hk+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 Hi in color i for some 1 ≤ i ≤ k + 1. We describe a general technique that supplies tight lower bounds for several numbers r(H1,H2, . . . ,Hk+1) when k ≥ 2, and the last graph Hk+1 is the complete graph Km 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(K3,K3,Km) = Θ(mpoly logm), thus solving (in a strong form) a conjecture of Erdős and Sós raised in 1979. Another special case of our result implies that r(C4, C4,Km) = Θ(mpoly logm) and that r(C4, C4, C4,Km) = Θ(m/ logm). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain estimates of character sums.