On the 3-Color Ramsey Numbers $R(C_4,C_4,W_{n})$

Abstract

For given graphs $G_1, G_2, \dots , G_k$, $k\ge 2$, the k-color Ramsey number, denoted by $R(G_1, G_2, \ldots , G_k)$, is the smallest integer N such that if we arbitrarily color the edges of a complete graph of order N with k colors, then it always contains a monochromatic copy of $G_i$ in color i, for some $1 \le i \le k$. Let $C_m$ be a cycle of length m and $W_{n}$ a wheel of order $n+1$. In this paper, we show that $R(C_4, C_4, W_{ n})\le n+\left\lceil \sqrt{4n+5}\right\rceil +3$ for $n=42, 48, 49, 50, 51, 52$ or $n\ge 56$. Furthermore, we prove that $R(C_4, C_4, W_{\ell ^2-\ell })\le \ell ^2+\ell +2$ for $\ell \ge 9$, and if $\ell$ is a prime power, then the equality holds.