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    New upper bound for multicolor Ramsey number of odd cycles

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    Let rk(C2m+1)r_k(C_{2m+1}) be the kk-color Ramsey number of an odd cycle C2m+1C_{2m+1} of length 2m+12m+1. It is shown that for each fixed mβ‰₯2m\ge2, rk(C2m+1)<ckk!r_k(C_{2m+1})<c^{k}\sqrt{k!} for all sufficiently large kk, where c=c(m)>0c=c(m)>0 is a constant. This improves an old result by Bondy and Erd\H{o}s (Ramsey numbers for cycles in graphs, J. Combin. Theory Ser. B 14 (1973) 46-54).Comment: 6 page
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