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 \(H_i \nsubseteq G_i\) for all \(1 \leq i \leq r\), then \(G\) is said to be \(r\)-colorable to \((H_1, H_2, \ldots, H_r)\). The multicolor Ramsey number \(R(H_1, H_2, \ldots, H_r)\) is the smallest integer \(n\) such that \(K_n\) is not \(r\)-colorable to \((H_1, H_2, \ldots, H_r)\). 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 \(m \geq 5\), which show that \(R(C_m, C_4, C_4) = m + 2\) for \(m \geq 11\).