On Ramsey Numbers $$R(K_4-e, K_t)$$

Abstract

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.