  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 $W_n$ denote a wheel with $n+1$ vertices. In 2014,
Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1},
W_{n})$ 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 $k$, $R(C_{2k+1}, W_{2j})=4j+1$. In 2015, Sanhueza-Matamala
confirmed this conjecture asymptotically, showing that $R(C_{2k+1}, W_{2j}) \le
4j+334$. In this paper, we prove the conjecture of Zhang, Zhang and Chen for
almost all of the remaining cases. In particular, we prove that
$R(C_{2k+1},W_{2j})=4j+1$ if $j-k \ge 251$, $k<j<3k/2$, and $j \ge 212299$.
