Ramsey numbers for a large tree versus multiple copies of complete graphs of different sizes

For two graphs G and H, let G ∪ H be the union of vertex-disjoint copy of G and H.And the Ramsey number R(G, H) is the minimum integer N such that any red-blue coloring of the edges of the complete graph K N contains either a red copy of G or a blue copy of H.If G is connected andwhere χ(H) is the chromatic number of H and s(H) is the size of the smallest color class taken over all proper vertex-colorings of H with χ(H) colors.Burr defined a connected graph G as H-good if the above inequality becomes equality.In this paper, for integers t ≥ 1 and m 1 ≥ m 2 ≥ • • • ≥ m t , we show that if n is sufficiently large, then any tree T n is t i=1 K mi -good.In particular, we show that the condition of n being sufficiently large can be relaxed when T n is a star.