On fan–wheel and tree–wheel Ramsey numbers

For 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, GG contains G1G1 as a subgraph or the complement of GG contains G2G2 as a subgraph. Let TnTn denote a tree of order nn, WnWn a wheel of order n+1n+1 and FnFn a fan of order 2n+12n+1. We establish Ramsey numbers for fans and trees versus wheels of even order, thereby extending several known results. In particular, we prove that R(Fn,Wm)=6n+1R(Fn,Wm)=6n+1 for odd m≥3m≥3 and n≥(5m+3)/4n≥(5m+3)/4, and that R(Tn,Wm)=3n−2R(Tn,Wm)=3n−2 for odd m≥3m≥3 and n≥m−2n≥m−2, and TnTn being a tree for which the Erdős–Sós Conjecture holds.