Rainbow Arithmetic Progressions

In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers n and k, the expression aw([n], k) denotes the smallest number of colors with which the integers {1, ... , n} can be colored and still guarantee there is a rainbow arithmetic progression of length k. We establish that aw([n], 3) = Theta(log n) and aw([n], k) = n(1-o(1)) for k >= 4.For positive integers n and k, the expression aw(Z(n), k) denotes the smallest number of colors with which elements of the cyclic group of order n can be colored and still guarantee there is a rainbow arithmetic progression of length k. In this setting, arithmetic progressions can "wrap around," and aw(Z(n), 3) behaves quite differently from aw([n], 3), depending on the divisibility of n. As shown in [Jungic et al., Combin. Prob. Comput., 2003], aw(Z(2m), 3) = 3 for any positive integer m. We establish that aw(Z(n), 3) can be computed from knowledge of aw(Z(p), 3) for all of the prime factors p of n. However, for k >= 4, the behavior is similar to the previous case, that is, aw(Z(n), k) = n(1-o(1)).