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.