On sequences without short zero-sum subsequences (Q6117239)

Let \((G,+,0)\) be a finite abelian group. A sequence over \(G\) is a finite unordered sequence with terms from \(G\) and repetition allowed. Let \[ S=g_1\cdot g_2\cdot\ldots\cdot g_{\ell}=\prod_{g\in G}g^{\mathsf v_g(S)} \] be a sequence over \(G\) of length \(\ell\). We denote \[ \mathsf h(S)=\max\{\mathsf v_g(S)\colon g\in G\} \] and \[ \Sigma(S):=\left\{\sum_{i\in I}g_i\colon \emptyset\neq I\subset [1,\ell]\right\}\,. \] We say \(S\) is zero-sum if \(\sigma(S):=g_1+\ldots+g_{\ell}=0\), and say a sequence \(T\) over \(G\) is a subsequence of \(S\) if \(\mathsf v_g(T)\le \mathsf v_g(S)\) for all \(g\in G\). It is well known that every sequence \(S\) over \(G\) of length not smaller than \(|G|\) has a nontrivial zero-sum subsequence of length not larger than \(\mathsf h(S)\). The associated inverse problem studied the structure of sequences without such a zero-sum subsequence. \textit{W. Gao} et al. [J. Comb. Theory, Ser. A 118, No. 2, 613--622 (2011; Zbl 1228.05303)] obtained that for every sequence \(S\) with \(\Sigma(S)\neq G\), if \(|\Sigma(S)|<2|S|-1\) and \(S\) has no nontrivial zero-sum subsequence of length \(\le \mathsf h(S)\), then there exists \(g\in G\) such that \(S=(n_1g)(n_2g)\cdot\ldots\cdot (n_{\ell}g)\) with \(1=n_1\le \ldots\le n_{\ell}\) and \(n_t\le \sum_{i=1}^{t-1}n_i\) for every \(t\in [2, \ell]\). In this manuscript, the authors took a step forward and obtained the structure of sequences \(S\) with \(|\Sigma(S)|=2|S|-1<|G|\) and without nontrivial zero-sum subsequence of length \(\le \mathsf h(S)\) (see Theorem 4), which can deduce some interesting corollaries (see Corollaries 5 and 6).