A small step forwards on the Erdős–Sós problem concerning the Ramsey numbers [formula omitted

Abstract

Let Δs=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 Δs. The best known concrete bounds on Δs are 3≤Δs≤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 Δs, but we still consider it a step towards to understanding its growth. We discuss some related questions and state two conjectures involving Δs, including the following: for some constant d and all s it holds that Δs−Δs+1≤d. We also prove that if the latter is true, then lims→∞Δs/s=0.