Rainbow arithmetic progressions

Anti-van der Waerden Numbers of 3-Term Arithmetic Progression

“…In this paper, we determine the exact value of aw([n], 3), which answers questions posed in [4] and confirms the following conjecture: Conjecture 1. [4] There exists a constant C such that aw([n], 3) ≤ ⌈log 3 n⌉ + C, for all n ≥ 3.…”

“…In [4,Theorem 1.6] it is shown that 3 ≤ aw(Z p , 3) ≤ 4 for every prime number p and that if aw(Z p , 3) = 4 then p ≥ 17. Furthermore, it is shown that the value of aw(Z n , 3) is determined by the values of aw(Z p , 3) for the prime factors p of n. We have included this theorem below with some notation change.…”

“…Theorem 3. [4] Let n be a positive integer with prime decomposition n = 2 e 0 p e 1 1 p e 2 2 • • • p es s for e i ≥ 0, i = 0, . .…”

“…in [4]. It is proved in [4] that for k ≥ 4, aw([n], k) = n 1−o (1) and aw(Z n , k) = n 1−o (1) . These results are obtained using results of Behrend [3] and Gowers [5] on the size of a subset of [n] with no k-AP.…”

Rainbow arithmetic progressions in finite abelian groups

“…Butler et. al.,in [3], proved upper and lower bounds for anti-van der Waerden numbers of [n] and Z n for k-APs, for 3 ≤ k.…”