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What is a least integer upper bound on van der Waerden number $W(r, k)$ amongthe powers of the integer $r$? We show how this can be found by expanding theinteger $W(r, k)$ into powers of $r$. Doing this enables us to find both aleast upper bound and a greatest lower bound on $W(r, k)$ that are some powersof $r$ 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 aconjecture posed by R. Graham is true, following immediately as a particularcase of the overall result.

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 $r$