On the Multi-Colored Ramsey Numbers of Paths and Even Cycles

In this paper we improve the upper bound on the multi-color Ramsey numbers of paths and even cycles. More precisely, we prove the following. For every $r\geq 2$ there exists an $n_0=n_0(r)$ such that for $n\geq n_0$ we have $$R_r(P_n) \leq \left( r - \frac{r}{16r^3+1} \right) n.$$ For every $r\geq 2$ and even $n$ we have $$R_r(C_n) \leq \left( r - \frac{r}{16r^3+1} \right) n + o(n) \text{ as }n\rightarrow \infty.$$ The main tool is a stability version of the Erdős-Gallai theorem that may be of independent interest.