The Ramsey numbers of large cycles versus small wheels

For two given graphs G and H, the Ramsey number R(G;H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we determine the Ramsey number R(Cn;Wm) for m = 4 and m = 5. We show that R(Cn;W4) = 2n ¡ 1 and R(Cn;W5) = 3n ¡ 2 for n ¸ 5. For larger wheels it remains an open problem to determine R(Cn;Wm).