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 \(2k \geq r+1\) and \(r\) is a prime power, we show that \(R_r(C_{2k}) > {r^2+2k-r-1}\).