The Ramsey numbers for disjoint unions of graphs

For given graphs GG and H,H, the Ramsey number R(G,H)R(G,H) is the smallest natural number nn such that for every graph FF of order nn: either FF contains GG or the complement of FF contains H.H. In this paper we investigate the Ramsey number of a disjoint union of graphs R(⋃i=1kGi,H). For any natural integer k  , we contain a general upper bound, R(kG,H)⩽R(G,H)+(k-1)|V(G)|R(kG,H)⩽R(G,H)+(k-1)|V(G)|. We also show that if m=2n-4m=2n-4, 2n-82n-8 or 2n-62n-6, then R(kSn,Wm)=R(Sn,Wm)+(k-1)nR(kSn,Wm)=R(Sn,Wm)+(k-1)n. Furthermore, if |Gi|>(|Gi|-|Gi+1|)(χ(H)-1)|Gi|>(|Gi|-|Gi+1|)(χ(H)-1) and R(Gi,H)=(χ(H)-1)(|Gi|-1)+1R(Gi,H)=(χ(H)-1)(|Gi|-1)+1, for each ii, then R(⋃i=1kGi,H)=R(Gk,H)+∑i=1k-1|Gi|.