On the Olson and the strong Davenport constants (Q449721)

Let \(G\) be a finite abelian group. Define \(O(G)\) as the least \(n\), such that every set \(S\subseteq G\) with \(|S|=n\) contains a non-empty subset \(T\) adding up to 0, and \(SD(G)\) largest \(n\), such that there exists a set \(S\) which does not contain a non-empty subset \(T\neq S\) adding up to 0. We clearly have \(O(G)-1\leq SD(G)\leq O(G)\), and equalities can occur on both sides. In the present work lower bounds for these invariants are found, which allow for the precise determination of \(O(G)\) and \(SD(G)\) for a large number of groups.NEWLINENEWLINESince the notion of zero-sum free sets does not behave well under group homomorphisms, the authors first generalize \(SD\) to measure the size of zero-sum free sequences with not too much repetition. They then give recursive lower bounds involving specific subgroups. These bounds require tricky constructions. Although these results are too technical to be stated here, they are quite general and shall certainly be applied to a variety of other problems as well.NEWLINENEWLINEAs applications the authors show that if \(G=\bigoplus_{i=1}^t \mathbb{Z}_{n_i}\) with \(n_1|n_2|\dots|n_t\), then \(SD(G)-\sum_{i=1}^t (n_i-1)\) can become arbitrary large, which is certainly striking. They also determine \(SD(G)\) for \(G=\mathbb{Z}_p, \mathbb{Z}_p^2\), \(p\) a sufficiently large prime, and all groups of exponent \(\leq 5\).