  We study the structure of red-blue edge colorings of complete graphs, with no
copies of the $n$-cycle $C_n$ in red, and no copies of the $n$-wheel $W_n = C_n
\ast K_1$ in blue, for an odd integer $n$. Our first main result is that in any
such coloring, deleting at most two vertices we obtain a vertex-partition of
$G$ into three sets such that the edges inside the partition classes are red,
and edges between partition classes are blue. As a second result, we obtain
bounds for the Ramsey numbers of $r(C_{2k+1},W_{2j})$ for $k < j$ integers,
which asymptotically confirm the values of $4j+1$, as it were conjectured by
Zhang et al.
