Some New Exact van der Waerden Numbers

TL;DR

This paper reports new exact values for van der Waerden numbers and provides a formula for the van der Waerden function in cases with a single differing parameter, advancing understanding of these combinatorial constants.

Contribution

It presents several newly determined exact van der Waerden numbers and a formula for the van der Waerden function when only one parameter differs from 2.

Findings

Several new exact van der Waerden numbers identified.

A precise formula for the van der Waerden function in specific cases derived.

Enhanced understanding of the behavior of van der Waerden numbers in structured scenarios.

Abstract

For positive integers $r,k_0,k_1,...,k_{r-1},$ the van der Waerden number $w(k_0,k_1,...,k_{r-1})$ is the least positive integer $n$ such that whenever $\{1,2,...,n\}$ is partitioned into $r$ sets $S_{0},S_{1},...,S_{r-1}$, there is some $i$ so that $S_i$ contains a $k_i$-term arithmetic progression. We find several new exact values of $w(k_0,k_1,...,k_{r-1})$. In addition, for the situation in which only one value of $k_i$ differs from 2, we give a precise formula for the van der Waerden function (provided this one value of $k_i$ is not too small)