A dynamical argument for a Ramsey property

TL;DR

The paper presents a dynamical approach to establish a new Ramsey property involving a super-polynomial function and the opposite-Ramsey number, revealing novel combinatorial insights.

Contribution

It introduces a dynamical argument to prove the existence of a super-polynomial function related to the opposite-Ramsey number, advancing combinatorial theory.

Findings

Existence of a super-polynomial function q(n) with specific properties.

Finite lim inf of the opposite-Ramsey number for certain parameters.

New dynamical method applied to Ramsey theory.

Abstract

We show by a dynamical argument that there is a positive integer valued function $q$ defined on positive integer set $\mathbb N$ such that $q([\log n]+1)$ is a super-polynomial with respect to positive $n$ and \[\liminf_{n\rightarrow\infty} r\left((2n+1)^2, q(n)\right)<\infty,\] where $r(\ ,\ )$ is the opposite-Ramsey number function.