A Radix Representation for each van der Waerden number $W(r, k)$ with $r$ colors: Why $\log_{r}W(r, k) < k^{2}$ is true whenever $k$ is the number of terms in the arithmetic progression

Abstract

Here we show that by expressing a van der Waerden number $W(r, k)$ by its radix polynomial representation, it not only is possible to locate each proper subset on $\mathbb{R}$ in which the van der Waerden number lies, but also to show that conditions exist for which the logarithm of the van der Waerden number necessarily is bounded above by the square of the number of terms $k$ in the arithmetic progression. Furthermore we also use the method to find a mathematical expression or formula for the ratio of two "consecutive" van der Waerden numbers of the kind $W(r, k)$, $W(r, k + 1)$.