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) = Θ(log n) and aw([n], k) = n1−o(1) for k ≥ 4. For positive integers n and k, the expression aw(Zn, 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(Zn, 3) behaves quite differently from aw([n], 3), depending on the divisibility of n. As shown in [Jungić et al., Combin. Probab. Comput., 2003], aw(Z2m , 3) = 3 for any positive integer m. We establish that aw(Zn, 3) can be computed from knowledge of aw(Zp, 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(Zn, k) = n1−o(1).