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On some developments of the Erdős-Ginzburg-Ziv theorem. II. (English) Zbl 1069.11007

Summary: Let \(S\) be a sequence of elements from the cyclic group \({\mathbb Z}_m\). We say \(S\) is zsf (zero-sum free) if there does not exist an \(m\)-term subsequence of \(S\) whose sum is zero. Denote by \(g(m,k)\) the least integer such that every sequence \(S\) with at least \(k\) distinct elements and length \(g(m,k)\) must contain an \(m\)-term subsequence whose sum is zero. Furthermore, denote by \(E(m,s)\) the set of all equivalence classes of zsf sequences with length \(s\), up to order and affine transformation, that are not a proper subsequence of another zsf sequence. In this paper, we first find for a sequence \(S\) of sufficient length, \(| S| \geq 2m-\lfloor m/4 \rfloor-2\), necessary and sufficient conditions in terms of a system of inequalities over the integers for \(S\) to be zsf. Among the consequences, we determine \(g(m,k)\) for large \(m\), namely \(g(m,k)=2m-\lfloor (k^2-2k+5)/4 \rfloor\) provided \(m \geq k^2-2k-3\), which in turn resolves two conjectures of the first and fourth authors. Next, using independent methods, we evaluate \(g(m,5)\) for every \(m \geq 5\). We conclude with the list of \(E(m,s)\) for every \(m\) and \(s\) satisfying \(2m-2 \geq s \geq \max\{2m-8, 2m-\lfloor m/4 \rfloor-2\}\).
Part I, cf. Colloq. Math. Soc. János Bolyai 60, 97–117 (1992; Zbl 1042.11510).

MSC:

11B50 Sequences (mod \(m\))
11B75 Other combinatorial number theory

Citations:

Zbl 1042.11510
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