The Structure of Maximal Zero-sum Free Sequences

Consider multisets A in the group G = (Z/nZ) such that no non-empty subset has sum zero. It is known for long that the maximal cardinality of such a set is 2n − 2, and there is a conjecture of Gao and Geroldinger describing the structure of such sets of maximal cardinality (called “property B”). Recently, Gao, Geroldinger and Grynkiewicz showed that it is enough to prove the conjecture for n prime. In the present article, we attack this prime case by making a case distinction on the largest multiplicities of elements occuring in A. The main result is a proof of the conjecture under the assumption that at least one of the two largest multiplicities is either sufficiently large or sufficiently small. For the cases where n is small, computers have been used in the proof.