Two Upper Bounds for both the van der Waerden Numbers $W(r, k + 1)$ and $W(r + 1, k)$, that show the Existence of a Recurrence Relation

TL;DR

This paper establishes new upper bounds for van der Waerden numbers using a finite power series expansion, demonstrating a recurrence relation that advances understanding of these combinatorial constants.

Contribution

It introduces a novel method employing radix polynomial representation to derive upper bounds and recurrence relations for van der Waerden numbers.

Findings

Derived upper bounds for van der Waerden numbers

Established recurrence relations for these bounds

Enhanced understanding of the structure of van der Waerden numbers

Abstract

Using a method we have utilized previously, namely through a finite power series expansion which also sometimes is known as the "radix polynomial" representation of an integer, we find an upper bound for a van der Waerden number that has a recurrence property.