Ramsey numbers of stars versus wheels of similar sizes

We study the Ramsey number R(W\"m,S\"n) for a star S\"n on n vertices and a wheel W\"m on m+1 vertices. We show that the Ramsey number R(W\"m,S\"n)=3n-2 for n=m,m+1, and m+2, where m=7 and odd. In addition, we give the following lower bound for R(W\"m,S\"n) where m is even: R(W\"m,S\"n)=2n+1 for all n=m=6.