On zero-sum subsequences of length [formula omitted] II

Abstract

Let G be an additive finite abelian group of exponent exp⁡(G). For every positive integer k, let skexp⁡(G)(G) denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length kexp⁡(G). Let ηkexp⁡(G)(G) denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length between 1 and kexp⁡(G). It is conjectured by Gao et al. that skexp⁡(G)(G)=ηkexp⁡(G)(G)+kexp⁡(G)−1 for all pairs of (G,k). This conjecture is a common generalization of several previous conjectures and has been confirmed for some special pairs of (G,k). In this paper we shall prove this conjecture for more pairs of (G,k). We also study the inverse problem associated with skexp⁡(G)(G), i.e., we determine the structure of sequences S of length skexp⁡(G)(G)−1 that have no zero-sum subsequence of length kexp⁡(G).