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 $\varepsilon$

TL;DR

This paper provides a simple proof establishing that $r^{k^{2}}$ is an upper bound for van der Waerden numbers $W(r, k)$ for sufficiently large $k$, using elementary divisibility properties instead of complex combinatorial methods.

Contribution

The paper introduces a novel, elementary proof that bounds van der Waerden numbers without relying on double induction or advanced combinatorial techniques.

Findings

Proves $r^{k^{2}}$ bounds $W(r, k)$ for large $k$

Uses divisibility properties rather than combinatorial arguments

Accessible approach for non-specialists in combinatorial number theory

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.