Ramsey numbers of stars versus wheels of similar sizes

We study the Ramsey number R(Wm,Sn) for a star Sn on n vertices and a wheel Wm on m+1 vertices. We show that the Ramsey number R(Wm,Sn)=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(Wm,Sn) where m is even: R(Wm,Sn)⩾2n+1 for all n⩾m⩾6.