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)