A small step forwards on the Erdos-Sos problem concerning the Ramsey numbers R(3, k)

Let Delta(s) = R(K-3, K-s) - R(K-3, Ks-1), where R(G, H) is the Ramsey number of graphs G and H defined as the smallest n such that any edge coloring of K-n with two colors contains G in the first color or H in the second color. In 1980, Erdos and Sos posed some questions about the growth of Delta(s). The best known concrete bounds on ds are 3 <= Delta(s) <= s, and they have not been improved since the stating of the problem. In this paper we present some constructions, which imply in particular that R(K-3, K-s) >= R(K-3, Ks-1 - e) + 4, and R(3, Ks+t-1) >= R(3, Ks+1 - e) R(3, Kt+1 - e) - 5 for s, t >= 3. This does not improve the lower bound of 3 on Delta(s), but we still consider it a step towards to understanding its growth. We discuss some related questions and state two conjectures involving Delta(s), including the following: for some constant d and all s it holds that Delta(s) - Delta(s+1) <= d. We also prove that if the latter is true, then lim(s ->infinity) Delta(s)/s = 0. (C) 2016 Elsevier B.V. All rights reserved.