How to find the least upper bound on the van der Waerden Number $W(r, k)$ that is some integer Power of the coloring Integer $r$

Abstract

What is a least integer upper bound on van der Waerden number $W(r, k)$ among the powers of the integer $r$? We show how this can be found by expanding the integer $W(r, k)$ into powers of $r$. Doing this enables us to find both a least upper bound and a greatest lower bound on $W(r, k)$ that are some powers of $r$ and where the greatest lower bound is equal to or smaller than $W(r, k)$. A finite series expansion of each $W(r, k)$ into integer powers of $r$ then helps us to find also a greatest real lower bound on any $k$ for which a conjecture posed by R. Graham is true, following immediately as a particular case of the overall result.