New bounds on the Ramsey number $r(I_m, L_n)$

TL;DR

This paper establishes new bounds for the Ramsey numbers involving independent sets and transitive tournaments in oriented graphs, providing exact values for specific cases and asymptotic bounds for general parameters.

Contribution

The authors improve existing upper bounds on $r(I_m, L_3)$ and determine exact values for $r(I_4, L_3)$ and $r(I_5, L_3)$, advancing the understanding of these Ramsey numbers.

Findings

Proved $r(I_4, L_3) = 15$ and $r(I_5, L_3) = 23$.

Improved upper bound on $r(I_m, L_3)$ to $m^2 - m + 3$.

Established asymptotic bounds showing $r(I_m, L_3) o heta(m^2 / ext{log} m)$.

Abstract

We investigate the Ramsey numbers $r(I_m, L_n)$ which is the minimal natural number $k$ such that every oriented graph on $k$ vertices contains either an independent set of size $m$ or a transitive tournament on $n$ vertices. Apart from the finitary combinatorial interest, these Ramsey numbers are of interest to set theorists since it is known that $r(\omega m, n) = \omega r(I_m, L_n)$, where $\omega$ is the lowest transfinite ordinal number, and $r(\kappa m, n) = \kappa r(I_m, L_n)$ for all initial ordinals $\kappa$. Continuing the research by Bermond from 1974 who did show $r(I_3, L_3) = 9$, we prove $r(I_4, L_3) = 15$ and $r(I_5, L_3) = 23$. The upper bounds for both the estimates above are obtained by improving the upper bound of $m^2$ on $r(I_m, L_3)$ due to Larson and Mitchell (1997) to $m^2 - m + 3$. Additionally, we provide asymptotic upper bounds on $r(I_m, L_n)$ for all $n \geq 3$. In particular, we show that $r(I_m, L_3) \in \Theta(m^2 / \log m)$.