Rainbow Arithmetic Progressions in Finite Abelian Groups

For positive integers n and k, the anti-van der Waerden number of Zn, denoted by aw(Zn, 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(Zn, 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 unitary anti-van der Waerden number of a group is also defined and determined.