Two Zero-Sum Problems and Multiple Properties

Abstract

In this paper we consider the following open problems: Conjecture0.1.Let S be a sequence of 3n−3 elements in Cn⊕Cn. If S contains no nonempty zero-sum subsequence of length not exceeding n, then S consists of three distinct elements, each appearing n−1 times. Conjecture0.2.Let S be a sequence of 4n−4 elements in Cn⊕Cn. If S contains no zero-sum subsequence of length n, then S consists of four distinct elements, each appearing n−1 times. We show that both Conjecture 0.1 and Conjecture 0.2 are multiplicative, i.e., if Conjecture 0.1 (Conjecture 0.2) holds both for n=k and n=l then it holds also for n=kl.