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\begin{document}$$G_1, G_2, \ldots , G_t$$\end{document} be given graphs. By G→(G1,G2,…,Gt)\documentclass[12pt]{minimal}
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\begin{document}$$G\rightarrow (G_1, G_2, \ldots , G_t)$$\end{document}, we mean if the edges of G are arbitrarily colored by t colors, then for some i, 1≤i≤t\documentclass[12pt]{minimal}
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\begin{document}$$1\le i\le t$$\end{document}, the spanning subgraph of G whose edges are colored with the i-th color, contains a copy of Gi\documentclass[12pt]{minimal}
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\begin{document}$$G_i$$\end{document}. The Ramsey number R(G1,G2,…,Gt)\documentclass[12pt]{minimal}
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\begin{document}$$R(G_1, G_2, \ldots , G_t)$$\end{document} is the smallest positive integer n such that Kn→(G1,G2,…,Gt)\documentclass[12pt]{minimal}
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\begin{document}$$K_n\rightarrow (G_1, G_2, \ldots , G_t)$$\end{document}, and the size Ramsey number R^(G1,G2,…,Gt)\documentclass[12pt]{minimal}
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\begin{document}$${\hat{R}}(G_1, G_2, \ldots , G_t)$$\end{document} is defined as min{|E(G)|:G→(G1,G2,…,Gt)}\documentclass[12pt]{minimal}
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\begin{document}$$\min \{|E(G)|:~G\rightarrow (G_1, G_2, \ldots , G_t)\}$$\end{document}. Also, for given graphs G1,G2,…,Gt\documentclass[12pt]{minimal}
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\begin{document}$$G_1, G_2, \ldots , G_t$$\end{document} with r=R(G1,G2,…,Gt)\documentclass[12pt]{minimal}
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\begin{document}$$r=R(G_1, G_2, \ldots , G_t)$$\end{document}, the star-critical Ramsey number R∗(G1,G2,…,Gt)\documentclass[12pt]{minimal}
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\begin{document}$$R_*(G_1, G_2, \ldots , G_t)$$\end{document} is defined as min{δ(G):G⊆Kr,G→(G1,G2,…,Gt)}\documentclass[12pt]{minimal}
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\begin{document}$$\min \{\delta (G):~G\subseteq K_r, ~G\rightarrow (G_1, G_2, \ldots , G_t)\}$$\end{document}. In this paper, the Ramsey number and also the star-critical Ramsey number of a forest versus any number of complete graphs will be computed exactly in terms of the Ramsey number of the complete graphs. As a result, the computed star-critical Ramsey number is used to give a tight bound for the size Ramsey number of a forest versus a complete graph.
