Let G and H be finite undirected graphs. The Ramsey number R(G, H) is the smallest integer n such that for every graph F of order n, either F contains a subgraph isomorphic to G or its complement F¯\documentclass[12pt]{minimal}
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\begin{document}$${\overline{F}}$$\end{document} contains a subgraph isomorphic to H. An (s, t)-graph is a graph that contains neither a clique of order s nor an independent set of order t. In this paper we obtain some inequalities involving Ramsey numbers of the form R(K4-e,Kt)\documentclass[12pt]{minimal}
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\begin{document}$$R(K_4-e,K_t)$$\end{document}. In particular, a constructive proof implies that if G is a (k,s+1)\documentclass[12pt]{minimal}
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\begin{document}$$(k,s+1)$$\end{document}-graph, H is a (k,t+1)\documentclass[12pt]{minimal}
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\begin{document}$$(k,t+1)$$\end{document}-graph, and both G and H contain a (Kk-e)\documentclass[12pt]{minimal}
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\begin{document}$$(K_k-e)$$\end{document}-free graph M as an induced subgraph, then we have R(Kk+1-e,Ks+t+1)>|V(G)|+|V(H)|+|V(M)|.\documentclass[12pt]{minimal}
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\begin{document}$$R(K_{k+1}-e,K_ {s+t+1}) > |V(G)| + |V(H)| + |V(M)|.$$\end{document} Furthermore, if s≤t\documentclass[12pt]{minimal}
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\begin{document}$$s \le t$$\end{document}, then R(K4-e,Ks+t+1)≥R(3,s+1)+R(3,t+1)+s\documentclass[12pt]{minimal}
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\begin{document}$$R(K_4-e,K_ {s+t+1}) \ge R(3,s+1)+R(3,t+1)+s$$\end{document}. In the experimental part, we use the (K4-e)\documentclass[12pt]{minimal}
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\begin{document}$$(K_4-e)$$\end{document}-free graph generation process to construct graphs witnessing lower bounds for R(K4-e,Kt)\documentclass[12pt]{minimal}
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\begin{document}$$R(K_4-e,K_t)$$\end{document}, and compare the results obtained by this approach to the results obtained by analogous triangle-free process. Finally, some open problems involving Ramsey numbers of the form R(K4-e,Kt)\documentclass[12pt]{minimal}
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\begin{document}$$R(K_4-e,K_t)$$\end{document} and their asymptotics are posed.
