A simple Proof that $r^{k^{2}}$ is an upper Bound on each van der Waerden Number $W(r, k)$, for each $k$ bounded below by a certain positive real Number $\varepsilon$

Abstract

Here we answer a conjecture by Ron Graham about getting finer upper bounds for van der Waerden numbers in the affirmative, but without the application of double induction or combinatorics as applied to sets of integers that contain some van der Waerden number as an element. Rather we obtain the result solely by exploiting certain properties of any integer greater than one that is divisible by another integer. Our mathematical methods are easily accessible by those whose field of specialization lies outside of combinatorial number theory, such as discrete mathematics, elementary number theory or analytic number theory.