The Asymptotic Behaviors of $\log_{r}W(r, k)$ and $\log_{k}W(r, k)$, when $W(r, k)$ is a van der Waerden Number

Abstract

We derive the asymptotic behaviors of $\log_{r}W(r, k)$ and $\log_{k}W(r, k)$, when $W(r, k)$ is a van der Waerden Number. We use the approach to consider the subsets on the real line in which $W(2, 7)$ might lie.