Rainbow arithmetic progressions in finite abelian groups

For positive integers $n$ and $k$, the anti-van der Waerden number of $\mathbb{Z}_n$, denoted by $\mathrm{aw}(\mathbb{Z}_n, k)$, is the minimum number of colors needed to color the elements of the cyclic group of order $n$ and guarantee there is a rainbow arithmetic progression of length $k$. Butler et al. showed a reduction formula for $\mathrm{aw}(\mathbb{Z}_n, 3)$ in terms of the prime divisors of $n$. In this paper, we analagously define the anti-van der Waerden number of a finite abelian group $G$ and show $\mathrm{aw}(G, 3)s$ is determined by the order of $G$ and the number of groups with even order in a direct sum isomorphic to $G$. The unitary anti-van der Waerden number of a group is also defined and determined.

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Received 26 March 2016

Published 7 December 2018