Zero-sum Analogues of van der Waerden's Theorem on Arithmetic Progressions

TL;DR

This paper studies zero-sum variants of van der Waerden's theorem, establishing bounds and relationships for these numbers in the context of arithmetic progressions and colorings.

Contribution

It introduces and analyzes the zero-sum analogues of van der Waerden numbers, revealing new bounds and a reciprocity with classical van der Waerden numbers.

Findings

Derived bounds for $w_{\mathfrak{z}}(k;r)$

Established reciprocity between van der Waerden and zero-sum numbers

Explored mixed monochromatic/zero-sum progressions

Abstract

Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $w_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $\chi:[1,w_{\mathrm{\mathfrak{z}}}(k;r)] \rightarrow \{0,1,\dots,r-1\}$ admits a $k$-term arithmetic progression $a,a+d,\dots,a+(k-1)d$ with $\sum_{j=0}^{k-1} \chi(a+jd) \equiv 0 \,(\mathrm{mod }\,r)$. We investigate these numbers as well as a "mixed" monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and $w_{\mathrm{\mathfrak{z}}}(k;r)$.