Abstract:

The graph with vertices in GF(16), whose edges connect points having difference equal to a cube, which was known to be extremal for the Ramsey numbers R(3,3,3) and R(K_3, K_6e), is shown to be extremal for R(K_4e, K_6e). The proof is obtained by using computer algorithms to analyze the properties of the family of graphs having no K_4e and having no K_5e in the complement. It is also shown that there is a unique graph, up to graph isomorphism, which is extremal for R(K_4e, K_7e), viz., the strongly regular Schlafli graph on 27 vertices, which has an automorphism group of size 51840. This follows easily from the result that R(K_4e, K_6e) is 17. 