New upper bound for multicolor Ramsey number of odd cycles

TL;DR

This paper establishes a new upper bound for the multicolor Ramsey number of odd cycles, showing it grows at most exponentially with the number of colors and factorially with the square root, improving previous results.

Contribution

The paper provides a significantly improved upper bound for the multicolor Ramsey number of odd cycles, refining the growth rate estimate for large numbers of colors.

Findings

Upper bound: r_k(C_{2m+1}) < c^k * sqrt(k!) for large k

Improves previous bounds by Bondy and Erdős

Applicable for fixed m ≥ 2

Abstract

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).