For two graphs G and H, let r(G, H) and r∗(G,H)\documentclass[12pt]{minimal}
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\begin{document}$$r_*(G,H)$$\end{document} denote the Ramsey number and star-critical Ramsey number of G versus H, respectively. In 1996, Li and Rousseau proved that r(Km,Ft,n)=tn(m-1)+1\documentclass[12pt]{minimal}
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\begin{document}$$r(K_{m},F_{t,n})=tn(m-1)+1$$\end{document} for m≥3\documentclass[12pt]{minimal}
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\begin{document}$$m\ge 3$$\end{document} and sufficiently large n, where Ft,n=K1+nKt\documentclass[12pt]{minimal}
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\begin{document}$$F_{t,n}=K_{1}+nK_{t}$$\end{document}. Recently, Hao and Lin proved that r(K3,F3,n)=6n+1\documentclass[12pt]{minimal}
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\begin{document}$$r(K_{3},F_{3,n})=6n+1$$\end{document} for n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document} and r∗(K3,F3,n)=3n+3\documentclass[12pt]{minimal}
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\begin{document}$$r_{*}(K_{3},F_{3,n})=3n+3$$\end{document} for n≥4\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 4$$\end{document}. In this paper, we show that r(Km,sFt,n)=tn(m+s-2)+s\documentclass[12pt]{minimal}
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\begin{document}$$r(K_{m}, sF_{t,n})=tn(m+s-2)+s$$\end{document} for sufficiently large n and, in particular, r(K3,sFt,n)=tn(s+1)+s\documentclass[12pt]{minimal}
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\begin{document}$$r(K_{3}, sF_{t,n})=tn(s+1)+s$$\end{document} for t∈{3,4},n≥t\documentclass[12pt]{minimal}
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\begin{document}$$t\in \{3,4\},n\ge t$$\end{document} and s≥1\documentclass[12pt]{minimal}
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\begin{document}$$s\ge 1$$\end{document}. We also show that r∗(K3,F4,n)=4n+4\documentclass[12pt]{minimal}
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\begin{document}$$r_{*}(K_{3}, F_{4,n})=4n+4$$\end{document} for n≥4\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 4$$\end{document} and establish an upper bound for r(F2,m,Ft,n)\documentclass[12pt]{minimal}
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\begin{document}$$r(F_{2,m},F_{t,n})$$\end{document}.
