标题： 彩虹等差数列在有限阿贝尔群中 显示英文标题

标题： Rainbow Arithmetic Progressions in Finite Abelian Groups

摘要： 对于正整数$n$和$k$，\emph{反范德瓦尔登数}的$\mathbb{Z}_n$，记为$aw(\mathbb{Z}_n,k)$，是给阶为$n$的循环群的元素着色所需的最小颜色数，以确保存在一个长度为$k$的彩虹等差数列。 Butler 等 展示了$aw(\mathbb{Z}_{n},3) = 3$的约简公式，该公式以$n$的素数因子表示。 在本文中，我们类似地定义了有限交换群$G$的反范德瓦尔登数，并证明$aw(G,3)$由$G$的阶和与$G$同构的直和中具有偶数阶的群的数量确定。 群的\emph{单位反对称数}也进行了定义并得到了确定。 显示英文摘要

摘要： 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.