Rainbow Arithmetic Progressions in Finite Abelian Groups

Abstract

For positive integers $n$ and $k$, the \emph{anti-van der Waerden number} of ${\mathbb{Z}}_n$, denoted by $aw({\mathbb{Z}}_n,k)$, 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 $aw({\mathbb{Z}}_n,3)=3$ 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 $aw(G,3)$ is determined by the order of $G$ and the number of groups with even order in a direct sum isomorphic to $G$. The \emph{unitary anti-van der Waerden number} of a group is also defined and determined.