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.

arXiv comment27 pages, two Tables, no figures. Some slight modifications and two appendices added. More specifics and description about the two exponents $n$, $m$, added as a prelude to Theorem 5.1, Section 5

References in corpus (1)

• Some New Exact van der Waerden Numbers