Stability and Ramsey numbers for cycles and wheels

We study the structure of red–blue edge colorings of complete graphs, with no copies of the nn-cycle CnCn in red, and no copies of the mm-wheel Wm=Cm∗K1Wm=Cm∗K1 in blue. Our first result is that, if we take n=mn=m and nn odd, in any such coloring, one can delete at most two vertices to obtain a graph with a vertex-partition 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 r(C2k+1,W2j)r(C2k+1,W2j) for k