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

Abstract

Let $r$ and $k$ be positive integers with $r\mid k$. Denote by $w_{\mathfrak{z}}(k;r)$ the minimum integer such that every coloring $\chi :[1,w_{\mathfrak{z}}(k;r)]\to \{0,1,\dots ,r-1\}$ 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 $w_{\mathfrak{z}}(k;r)$.