SUMMARY
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=2r\geq 2 there exists an n0=n0(r)n_0=n_0(r) such that for n=n0n\geq n_0 we have Rr(Pn)=(r-r16r3+1)n.R_r(P_n) \leq \left( r - \frac{r}{16r^3+1} \right) n. For every r=2r\geq 2 and even nn we have Rr(Cn)=(r-r16r3+1)n+o(n) as n?8.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 Erdos-Gallai theorem that may be of independent interest.