Bounds on Van der Waerden Numbers and Some Related Functions

TL;DR

This paper establishes new bounds on Van der Waerden numbers, providing lower bounds for certain cases, upper bounds for others, and analyzing related functions to deepen understanding of arithmetic progression colorings.

Contribution

It introduces new lower bounds for w(k,m), provides bounds for specific cases like w(k,4) and w(4;s), and explores related functions in the context of Van der Waerden numbers.

Findings

Lower bounds for w(k,m) for fixed m

Upper bounds for w(k,4) and w(4;s)

Values of w(k,3) closely match the lower bound for 5 ≤ k ≤ 16

Abstract

For positive integers $s$ and $k_1, k_2, ..., k_s$, let $w(k_1,k_2,...,k_s)$ be the minimum integer $n$ such that any $s$-coloring $\{1,2,...,n\} \to \{1,2,...,s\}$ admits a $k_i$-term arithmetic progression of color $i$ for some $i$, $1 \leq i \leq s$. In the case when $k_1=k_2=...=k_s=k$ we simply write $w(k;s)$. That such a minimum integer exists follows from van der Waerden's theorem on arithmetic progressions. In the present paper we give a lower bound for $w(k,m)$ for each fixed $m$. We include a table with values of $w(k,3)$ which match this lower bound closely for $5 \leq k \leq 16$. We also give an upper bound for $w(k,4)$, an upper bound for $w(4;s)$, and a lower bound for $w(k;s)$ for an arbitrary fixed $k$. We discuss a number of other functions that are closely related to the van der Waerden function.