The Ramsey numbers of large star and large star-like trees versus odd 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 shall study the Ramsey number R(T_n, W_n) for a star-like tree T_n with n vertices and a wheel W_m with m + 1 vertices and m odd. We show that the Ramsey number R(S_n, W_m) = 3n - 2 for n ≥ 2m - 4, m ≥ 5 and m odd, where S_n denotes the star on n vertices. We conjecture that the Ramsey number is the same for general trees on n vertices, and support this conjecture by proving it for a number of star-like trees.