Granville, Andrew (ed.) et al., Additive combinatorics. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4351-2/pbk). CRM Proceedings and Lecture Notes 43, 307-326 (2007).
Let be a finite abelian group. The Davenport constant is the least integer such that any multiset consisting of elements of has a (non-empty) subset whose sum is 0. Write , where divides . Then one easily obtains a lower bound for the Davenport constant: . For groups of rank , one has equality. For groups of rank , there are infinitely examples with . Whether one always has equality in rank 3 is still open. The present article proves equality for the groups , with . (Before, the smallest unknown case was .)
The method used in the proof is an enhancement of the “inductive method” used by C. Delorme, O. Ordaz and D. Quiroz [Discrete Math. 237, No. 1–3, 119–128 (2001; Zbl 1003.20025)]. Using the projection , each multiset yields a multiset . Now suppose that is coprime to 3. (The case has already been treated by Delorme, Ordaz and Quiroz.) Then we have , and the information which gets lost by applying to can be encoded in a map . Using these notations, a subset has sum zero if and only if its projection has sum zero and . In this way, the question about multisets in has been translated into a question about multisets in with functions on them; this problem is easier to solve (but requires a lot of computation in ).