The Ramsey numbers for disjoint unions of trees

For given graphs G   and H,H, the Ramsey number  R(G,H)R(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F   contains H.H. In this paper, we investigate the Ramsey number R(∪G,H)R(∪G,H), where G is a tree and H   is a wheel WmWm or a complete graph KmKm. We show that if n⩾3n⩾3, then R(kSn,W4)=(k+1)nR(kSn,W4)=(k+1)n for k⩾2k⩾2, even n   and R(kSn,W4)=(k+1)n-1R(kSn,W4)=(k+1)n-1 for k⩾1k⩾1 and odd n  . We also show that R(⋃i=1kTni,Km)=R(Tnk,Km)+∑i=1k-1ni.