Result: On zero-sum subsequences of restricted size. IV

For a finite abelian group \(G\) the invariant \(s(G)\) (resp. the invariant \(s_0(G)\)) is the smallest integer \(\ell\in \mathbb N\) such that every sequence \(S\) in \(G\) of length \(| S| \geq \ell \) has a subsequence \(T\) with sum zero and length \(| T| =\exp(G)\) (resp. length \(| T| \equiv \bmod \exp(G)\)). The Davenport constant \(D(G)\) of \(G\) is the smallest integer \(\ell\in \mathbb N\) such that every sequences \(S\) in \(G\) with length \(| S| \geq \ell\) contains a zero-sum subsequence. For every \(n\in \mathbb N\) is denoted by \(C_n\) a cyclic group with \(n\) elements. The main result of this article is the following theorem: Let \(m,n\in \mathbb N\) with \(n\geq \frac 13 (m^2-m+1)\). If \(s_0(C^2_m)=3m-2\) and \(D(C^3_n)=3n-2\), then \(s_0(C^2_{mn})=3mn-2\). For the first three parts, see J. Number Theory 61, No. 1, 97--102 (1996; Zbl 0870.11016), Discrete Math. 271, No. 1-3, 51--59 (2003; Zbl 1089.11012) and Ars Comb. 61, 65--72 (2001; Zbl 1101.11311).