The Ramsey numbers for disjoint unions of trees

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For given graphs G and H , the Ramsey number 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 . In this paper, we investigate the Ramsey number R ( ∪ G , H ) , where G is a tree and H is a wheel W m or a complete graph K m . We show that if n ⩾ 3 , then R ( kS n , W 4 ) = ( k + 1 ) n for k ⩾ 2 , even n and R ( kS n , W 4 ) = ( k + 1 ) n - 1 for k ⩾ 1 and odd n. We also show that R ( ⋃ i = 1 k T n i , K m ) = R ( T n k , K m ) + ∑ i = 1 k - 1 n i .