The Asymptotic Behaviors of $\log_{r}W(r, k)$ and $\log_{k}W(r, k)$, when $W(r, k)$ is a van der Waerden Number

TL;DR

This paper investigates the asymptotic properties of logarithmic functions of van der Waerden numbers, providing insights into their growth and potential bounds, with implications for combinatorial number theory.

Contribution

It introduces new asymptotic analyses of $ ext{log}_r W(r,k)$ and $ ext{log}_k W(r,k)$ for van der Waerden numbers, advancing understanding of their behavior.

Findings

Derived asymptotic behaviors of the logarithmic functions

Estimated bounds for $W(2,7)$ on the real line

Provided a framework for analyzing van der Waerden numbers asymptotically

Abstract

We derive the asymptotic behaviors of $\log_{r}W(r, k)$ and $\log_{k}W(r, k)$, when $W(r, k)$ is a van der Waerden Number. We use the approach to consider the subsets on the real line in which $W(2, 7)$ might lie.