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 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.