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.
Mathematics Subject Classification (2000):05C55
Unable to display preview. Download preview PDF.