Ramsey numbers of trees versus fans

For two given graphs G1G1 and G2G2, the Ramsey number R(G1,G2)R(G1,G2) is the smallest integer NN such that, for any graph GG of order NN, either GG contains G1G1 as a subgraph or the complement of GG contains G2G2 as a subgraph. Let TnTn be a tree of order nn, SnSn a star of order nn, and FmFm a fan of order 2m+12m+1, i.e.,  mm triangles sharing exactly one vertex. In this paper, we prove that R(Tn,Fm)=2n−1R(Tn,Fm)=2n−1 for n≥3m2−2m−1n≥3m2−2m−1, and if Tn=SnTn=Sn, then the range can be replaced by n≥max{m(m−1)+1,6(m−1)}n≥max{m(m−1)+1,6(m−1)}, which is tight in some sense.