Abstract
For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that, for any graph G of order N, either G contains G1 as a subgraph or the complement of G contains G2 as a subgraph. Let Tn be a tree of order n, Sn a star of order n, and Fm a fan of order 2m+1, i.e., m triangles sharing exactly one vertex. In this paper, we prove that R(Tn,Fm)=2n−1 for n≥3m2−2m−1, and if Tn=Sn, then the range can be replaced by n≥max{m(m−1)+1,6(m−1)}, which is tight in some sense.
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