On the Erdös-Ginzburg-Ziv theorem
The authors present a further generalization of the Erdös-Ginzburg-Ziv theorem. Specifically, let \(n\) be a natural number, \(A\) be a set and let \(f:A\to \mathbb{Z}_n\). Erdös et al. proved that there exists an \(n\)-subset \(S\subseteq A\) such that \(\sum_{x\in S}f(x)=0\), when \(|A|\geq 2n-1\). Subsequently \textit{A. Bialostocki} and \textit{P. Dierker} [Discrete Math. 110, No. 1-3, 1-8 (1992; Zbl 0774.05065)] proved that for \(|A|=2n-2\) and for all \(f:A\to \mathbb{Z}_n\) such that \(|f(A)|\geq 3\) there exists an \(n\)-subset \(S\subseteq A\) such that \(\sum_{x\in S}f(x)=0\) or there are \(a,b\in \mathbb{Z}_n\) such that \(\mathbb{Z}_n\) is generated by \(b-a\) and \(|f^{-1}(a)|=|f^{-1}(b)|=n-1\). More recently \textit{A. Bialostocki} and \textit{M. Lotspeich} showed that for \(|A|=2n-3\) and \(|f(A)|\geq 4\), there exists an \(n\)-subset \(S\subseteq A\) such that \(\sum_{x\in S}f(x)=0\). The authors generalize the above results as follows. Let \(A\) be a set with cardinality \(2n-3\) and \(f:A\to \mathbb{Z}_n\). Then (1) there exists an \(n\)-subset \(S\subseteq A\) such that \(\sum_{x\in S}f(x)=0\), or (2) there are \(a,b\in \mathbb{Z}_n\) such that \(\mathbb{Z}_n\) is generated by \(b-a\) and \(|f^{-1}(a)|=n-1\) and one of the following conditions holds: (i) \(|A|\leq 2n-2\) and \(|f^{-1}(b)|=|A|-n+1\); (ii) \(|A|=2n-3\) and \(|f^{-1}(b)|=n-3\) and \(|f^{-1}(2b-a)|=1\).
scientific article; zbMATH DE number 897590