Year: 2026
Type: preprint
Abstract: For a nontrivial finite abelian group $G$, let $ν(G)$ be the smallest {nonnegative} integer $\ell$ such that every zero-sum free sequence $T$ over $G$ of length at least $\ell$ has the following property: all nonzero elements of $G$ that do not occur as a subsequence sum of $T$ lie in a proper coset of some subgroup of $G$. It is easy to check that $\mathsf d (G)-1 \le ν(G) \le \mathsf d (G)$, where $\mathsf d (G)$ is the small Davenport constant of $G$. A conjecture by Gao from the year 2000 stated that equality should always hold at the lower bound. This conjecture has since been confirmed for many families of groups (including all p-groups and groups of rank at most two). In this paper, we disprove the conjecture.
Source: arXiv (Cornell University)
Institutions: University of Graz, Nawi Graz
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213428853