Remarks on a generalization of the Davenport constant

A generalization of the Davenport constant is investigated. For a finite abelian group GG and a positive integer kk, let Dk(G) denote the smallest ℓℓ such that each sequence over GG of length at least ℓℓ has kk disjoint non-empty zero-sum subsequences. For general GG, expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence (Dk(G))k∈N is eventually an arithmetic progression with difference exp(G)exp(G), and several questions arising from this fact are investigated. For elementary 22-groups, Dk(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).