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 \({\overline{F}}\) 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 \(R(K_4-e,K_t)\). In particular, a constructive proof implies that if G is a \((k,s+1)\)-graph, H is a \((k,t+1)\)-graph, and both G and H contain a \((K_k-e)\)-free graph M as an induced subgraph, then we have \(R(K_{k+1}-e,K_ {s+t+1}) > |V(G)| + |V(H)| + |V(M)|.\) Furthermore, if \(s \le t\), then \(R(K_4-e,K_ {s+t+1}) \ge R(3,s+1)+R(3,t+1)+s\). In the experimental part, we use the \((K_4-e)\)-free graph generation process to construct graphs witnessing lower bounds for \(R(K_4-e,K_t)\), 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 \(R(K_4-e,K_t)\) and their asymptotics are posed.

令G和H为有限无向图。拉姆齐数R（G，  H）是最小的整数n，因此对于n阶的每个图F，F都包含与G同构的子图，或者其补\（{\ overline {F}} \）包含同构子图到^ h。（s，  t）图是既不包含s阶团也不包含t的独立集合的图。在本文中，我们获得了一些不等式，涉及形式为\（R（K_4-e，K_t）\）的Ramsey数。特别地，一个构造性证明意味着，如果G是\（（k，s + 1）\）-图，H是\（（k，t + 1）\）-图，并且G和H都包含一个\（（（K_k-e）\） -自由图M作为诱导子图，则我们有\（R（K_ {k + 1} -e，K_ {s + t + 1}）> | V（G）| + | V（H）| + | V（M）| .. \）此外，如果\（s \ le t \），则\（R（K_4-e，K_ {s + t + 1}）\ ge R （3，s + 1）+ R（3，t + 1）+ s \）。在实验部分，我们使用\（（K_4-e）\）无图生成过程，以构造\（R（K_4-e，K_t）\）下界的图，并将通过这种方法获得的结果与通过类似无三角形方法获得的结果进行比较。最后，提出了一些涉及形式为\（R（K_4-e，K_t）\）的Ramsey数及其渐近性的开放问题。