Zero-sum analogues of van der Waerden’s theorem on arithmetic progressions

Let $r$ and $k$ be positive integers with $r \vert k$. Denote by $z(k; r)$ the minimum integer such that every coloring $\chi : [ 1, z(k; r)] \to \lbrace 0, 1, \dotsc , r - 1 \rbrace$ admits a $k$-term arithmetic progression $a, a + d, \dotsc , a + (k - 1) d$ with $\sum^{k−1}_{j=0} \chi (a + jd) \equiv 0 (\operatorname{mod} r)$. 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 $z(k; r)$.

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Received 13 February 2018

Published 14 January 2020