New upper bound for multicolor Ramsey number of odd cycles

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 cycles in graphs, J. Combin. Theory Ser. B 14 (1973) 46-54).Comment: 6 page