Let G be an additive finite abelian group of exponent $\exp (G)$. For every positive integer k, let ${\mathsf{s}}_{k\exp (G)}(G)$ denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length $k\exp (G)$. Let ${\eta }_{k\exp (G)}(G)$ denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length between 1 and $k\exp (G)$. It is conjectured by Gao et al. that ${\mathsf{s}}_{k\exp (G)}(G)={\eta }_{k\exp (G)}(G)+k\exp (G)-1$ 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 ${\mathsf{s}}_{k\exp (G)}(G)$, i.e., we determine the structure of sequences S of length ${\mathsf{s}}_{k\exp (G)}(G)-1$ that have no zero-sum subsequence of length $k\exp (G)$.

令G为指数的可加有限阿贝尔群$\mathrm{经验值}(G)$. 对于每个正整数k，令${\mathsf{秒}}_{克\mathrm{经验值}(G)}(G)$表示最小整数t使得G上长度为t 的每个序列都有一个长度为零的子序列$克\mathrm{经验值}(G)$. 让${\eta }_{克\mathrm{经验值}(G)}(G)$表示最小整数t，使得G上长度为t 的每个序列都有一个长度在 1 和$克\mathrm{经验值}(G)$. 由 Gao 等人推测。那${\mathsf{秒}}_{克\mathrm{经验值}(G)}(G)={\eta }_{克\mathrm{经验值}(G)}(G)+克\mathrm{经验值}(G)-1$ 对于所有对 $(G,克)$. 这个猜想是对之前几个猜想的共同推广，并且已经被一些特殊的对$(G,克)$. 在本文中，我们将证明这个猜想的更多对$(G,克)$. 我们还研究了与${\mathsf{秒}}_{克\mathrm{经验值}(G)}(G)$, 即我们确定长度为S的序列的结构${\mathsf{秒}}_{克\mathrm{经验值}(G)}(G)-1$ 没有零和长度的子序列 $克\mathrm{经验值}(G)$.