The Ramsey numbers for disjoint union of trees versus W 4

The Ramsey number for a graph G versus a graph H, denoted by R(G, H), is the smallest positive integer n such that for any graph F of order n, either F contains G as a subgraph or F contains H as a subgraph. In this paper, we investigate the Ramsey numbers for union of trees versus small cycle and small wheel. We show that if ni is odd and 2ni+1 ≥ ni for every i, then R( Sk i=1 Tni , W4) = R(Tnk , W4) + Pk−1 i=1 ni for k ≥ 1.. Furthermore, we show that 1. If ni is even and 2ni+1 ≥ ni + 1 for every i, then R( Sk i=1 Sni , W4) = 2nk + Pk−1 i=1 ni for k ≥ 2, 2. If ni is odd and 2ni+1 ≥ ni for every i, then R( Sk i=1 Sni , W4) = R(Snk , W4) + Pk−1 i=1 ni for k ≥ 1.