SUMMARY
Burr, Erdos, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that R(Tn,Cm)=2n-1R(T_n, C_m) = 2n - 1 for all odd m=3m \ge 3 and n=756m10n \ge 756m^{10}, where TnT_n is a tree with nn vertices and CmC_m is an odd cycle of length mm. They proposed to study the minimum positive integer n0(m)n_0(m) such that this result holds for all n=n0(m)n \ge n_0(m), as a function of mm. In this paper, we show that n0(m)n_0(m) is at most linear. In particular, we prove that R(Tn,Cm)=2n-1R(T_n, C_m) = 2n - 1 for all odd m=3m \ge 3 and n=25mn \ge 25m. Combining this with a result of Faudree, Lawrence, Parsons and Schelp yields n0(m)n_0(m) is bounded between two linear functions, thus identifying n0(m)n_0(m) up to a constant factor.