A step forwards on the Erd\H{o}s-S\'os problem concerning the Ramsey numbers $R(3,k)$

TL;DR

This paper advances understanding of the growth of the difference in Ramsey numbers involving triangles and larger cliques, providing new constructions and conjectures about their behavior.

Contribution

It introduces new constructions that relate different Ramsey numbers and proposes conjectures on the boundedness of their differences, advancing the theoretical understanding of $R(3,k)$ growth.

Findings

Established a new lower bound relation for $R(K_3,K_s)$

Proposed conjectures on the boundedness of $ riangle_s$ differences

Discussed implications of these conjectures on the growth rate of $ riangle_s$

Abstract

Let $\Delta_s=R(K_3,K_s)-R(K_3,K_{s-1})$, 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, Erd\H{o}s and S\'{o}s posed some questions about the growth of $\Delta_s$. The best known concrete bounds on $\Delta_s$ are $3 \le \Delta_s \le s$, and they have not improved since the stating of the problem. In this paper we present some constructions, which imply in particular that $R(K_3,K_s) \ge R(K_3,K_{s-1}-e) + 4$. 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 $\Delta_s - \Delta_{s+1} \leq d$. We also prove that if the latter is true, then $\lim_{s \rightarrow \infty} \Delta_s/s=0$.