Ramsey Numbers for Complete Graphs Versus Generalized Fans

For two graphs G and H, let r(G, H) and r(*)(G, H) denote the Ramsey number and star-critical Ramsey number of G versus H, respectively. In 1996, Li and Rousseau proved that r(K-m, F-t,F-n) = tn(m - 1) + 1 for m >= 3 and sufficiently large n, where F-t,F-n = K-1 + nK(t). Recently, Hao and Lin proved that r(K-3, F-3,F-n) = 6n + 1 for n >= 3 and r(*)(K-3, F-3,F-n) = 3n + 3 for n >= 4. In this paper, we show that r(K-m, sF(t,n)) = tn(m + s - 2) + s for sufficiently large n and, in particular, r(K-3, sF(t,n)) = tn(s + 1) + s for t is an element of {3, 4}, n >= t and s >= 1. We also show that r*(K-3, F-4,F-n) = 4n + 4 for n >= 4 and establish an upper bound for r(F-2,F-m, F-t,F-n).