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

TL;DR

This paper introduces a radix polynomial representation of van der Waerden numbers to locate their bounds and demonstrates that their logarithm is bounded above by the square of the progression length, also deriving ratios of consecutive numbers.

Contribution

It presents a novel radix polynomial approach to analyze van der Waerden numbers and establishes bounds on their logarithm relative to the progression length.

Findings

Logarithm of $W(r, k)$ is bounded by $k^2$ under certain conditions.

Method to locate $W(r, k)$ within specific real intervals.

Derived formula for the ratio of consecutive van der Waerden numbers.

Abstract

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