A dynamical argument for a Ramsey property

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.