On the Multi-Colored Ramsey Numbers of Paths and Even Cycles
Keywords:
Ramsey numbers, Paths, Cycles
Abstract
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≥2 there exists an n0=n0(r) such that for n≥n0 we have Rr(Pn)≤(r−r16r3+1)n. For every r≥2 and even n we have Rr(Cn)≤(r−r16r3+1)n+o(n) as n→∞. The main tool is a stability version of the Erdős-Gallai theorem that may be of independent interest.