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

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