Rainbow Arithmetic Progressions in Finite Abelian Groups

TL;DR

This paper extends the concept of anti-van der Waerden numbers to finite abelian groups, providing formulas based on group order and structure, including the unitary variant.

Contribution

It generalizes anti-van der Waerden numbers from cyclic groups to all finite abelian groups and introduces the unitary version, with explicit formulas.

Findings

$aw(G,3)$ depends on the order of $G$ and the number of even-order components

Formulas for $aw(G,3)$ are established for finite abelian groups

The unitary anti-van der Waerden number is characterized

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.