A dynamical argument for a Ramsey property

We show by a dynamical argument that there is a positive integer valued function $q$ defined on positive integer set $\mathbb{N}$ such that $q([\log n]+1)$isasuper-polynomialwithrespecttopositive$n$ and $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}r((2n+1{)}^2,q(n))<\mathrm{\infty },$$ where $r(,)$ is the opposite-Ramsey number function.