The Ramsey Numbers $r(m{C}_4,n{C}_5)$

Abstract

If $G$ and $H$ are graphs, define the Ramsey number $r(G,H)$ to be the least number $p$ such that if the edges of the complete graph $K_p$ are colored red and blue (say), either the red graph contains a copy of $G$, or the blue graph contains a copy of $H$. In this paper, we determine the Ramsey number $r(m{C}_4,n{C}_5)$ for any $m\ge 1,n\ge 1$.