Ramsey numbers of odd cycles versus larger even wheels

The generalized Ramsey number R(G(1), G(2)) is the smallest positive integer N such that any red-blue coloring of the edges of the complete graph K-N either contains a red copy of G(1) or a blue copy of G(2). Let C-m denote a cycle of length m and W, denote a wheel with n+1 vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers R(C2k+1 W-n) of odd cycles versus larger wheels, leaving open the particular case where n = 2j is even and k < j < 3k/2. They conjectured that for these values of j and k, R(C2k+1, W-2j) = 4j + 1. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that R(C2k+1, W-2j) <= 4j + 334. In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that R(C2k+1, W-2j) = 4j+1 if j - k >= 251, k < j < 3k/2, and j >= 212299. (C) 2018 Elsevier B.V. All rights reserved.