On Multicolor Ramsey Numbers for Even Cycles in Graphs

Abstract

The multicolor Ramsey number $R_r(H)$ is defined to be the smallest integer $n=n(r)$ with the property that any $r$-coloring of the edges of complete graph $K_n$ must result in a monochromatic subgraph of $K_n$ isomorphic to $H$. In this paper, we study the case that $H$ is a cycle of length $2k$. If $2k\ge r+1$ and $r$ is a prime power, we show that $R_r(C_{2k})>r^2+2k-r-1$.