On the Three Color Ramsey Numbers $R(C_m,C_4,C_4)$

Abstract

Let $G_i$ be the subgraph of $G$ whose edges are in the $i$-th color in an $r$-coloring of the edges of $G$. If there exists an $r$-coloring of the edges of $G$ such that $H_i⊈G_i$ for all $1\le i\le r$, then $G$ is said to be $r$-colorable to $(H_1,H_2,\dots ,H_r)$. The multicolor Ramsey number $R(H_1,H_2,\dots ,H_r)$ is the smallest integer $n$ such that $K_n$ is not $r$-colorable to $(H_1,H_2,\dots ,H_r)$. It is well known that $R(C_m,C_4,C_4)=m+2$ for sufficiently large $m$. In this paper, we determine the values of $R(C_m,C_4,C_4)$ for $m\ge 5$, which show that $R(C_m,C_4,C_4)=m+2$ for $m\ge 11$.