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

Zbl 1042.11510
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