A generalization of the Davenport constant is investigated. For a finite abelian group G and a positive integer k, let D k (G) denote the smallest ℓ such that each sequence over G of length at least ℓ has k disjoint non-empty zero-sum subsequences. For general G, expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence (D k (G)) k∈N is eventually an arithmetic progression with difference exp(G), and several questions arising from this fact are investigated. For elementary 2-groups, D k (G) is investigated in detail; in particular, the exact values are determined for groups of rank four and five (for rank at most three they were already known).
Let G be an additive, finite abelian group. The critical number cr(G) of G is the smallest positive integer such that for every subset S ⊂ G \ {0} with |S| the following holds: Every element of G can be written as a nonempty sum of distinct elements from S. The critical number was first studied by P. Erdős and H. Heilbronn in 1964, and due to the contributions of many authors the value of cr(G) is known for all finite abelian groups G except for G ∼ = Z/pqZ where p, q are primes such that p + 2 √ p − 2 + 1 < q < 2p. We determine that cr(G) = p + q − 2 for such groups.
Let H be a Krull monoid such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). For k ∈ N let V k (H) denote the set of all m ∈ N with the following property : There exist atoms (irreducible elements)We show that the sets V k (H) are intervals for all k ∈ N. This solves Problem 37 in [4].
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