Ramsey numbers of odd cycles versus larger even wheels

Abstract

The generalized Ramsey number R(G1,G2) is the smallest positive integer N such that any red–blue coloring of the edges of the complete graph KN either contains a red copy of G1 or a blue copy of G2. Let Cm denote a cycle of length m and Wn denote a wheel with n+1 vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers R(C2k+1,Wn) 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,W2j)=4j+1. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that R(C2k+1,W2j)≤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,W2j)=4j+1 if j−k≥251, k<j<3k∕2, and j≥212299.