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Let $r_k(C_{2m+1})$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$of length $2m+1$. It is shown that for each fixed $m\ge2$,\[r_k(C_{2m+1})<c^{k}\sqrt{k!}\] for all sufficiently large $k$, where$c=c(m)>0$ is a constant.  This improves an old result by Bondy and Erd\H{o}s (Ramsey numbers for cyclesin graphs, J. Combin. Theory Ser. B 14 (1973) 46-54).

New upper bound for multicolor Ramsey number of odd cycles