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

Let Δ s = R ( K 3 , K s ) − R ( K 3 , K s − 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, Erdős and Sós posed some questions about the growth of Δ s . The best known concrete bounds on Δ s are 3 ≤ Δ 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 , K s − 1 − e ) + 4 , and R ( 3 , K s + t − 1 ) ≥ R ( 3 , K s + 1 − e ) + R ( 3 , K t + 1 − e ) − 5 for s , t ≥ 3 . This does not improve the lower bound of 3 on Δ s , but we still consider it a step towards to understanding its growth. We discuss some related questions and state two conjectures involving Δ s , including the following: for some constant d and all s it holds that Δ s − Δ s + 1 ≤ d . We also prove that if the latter is true, then lim s → ∞ Δ s / s = 0 . [ABSTRACT FROM AUTHOR]