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

Let ∆s = R(K3,Ks) − R(K3,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 Kn with two colors contains G in the first color or H in the second color. In 1980, Erdős and Sós posed some questions about the growth of ∆s. The best known concrete bounds on ∆s are 3 ≤ ∆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(K3,Ks) ≥ R(K3,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 ∆s, but we still consider it a step towards to understanding its growth. We discuss some related questions and state two conjectures involving ∆s, including the following: for some constant d and all s it holds that ∆s −∆s+1 ≤ d. We also prove that if the latter is true, then lims→∞∆s/s = 0.