The structure of maximal zerosum free Sequences
Abstract
Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zerosum free, if no subsequence has sum zero. It is known that the maximal length of such a zerosum free sequence is 2n2, and Gao and Geroldinger conjectured that every zerosum free sequence of this length contains an element with multiplicity at least n2. By recent results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify the conjecture for n prime. Now fix a sequence (a_i) of length 2n2 with maximal multiplicity of elements at most n3. There are different approeaches to show that (a_i) contains a zerosum; some work well when (a_i) does contain elements with high multiplicity, others work well when all multiplicities are small. The aim of this article is to initiate a systematic approach to property B via the highest occurring multiplicities. Our main results are the following: denote by m_1 >= m_2 the two maximal multiplicities of (a_i), and suppose that n is sufficiently big and prime. Then (a_i) contains a zerosum in any of the following cases: when m_2 >= 2/3n, when m_1 > (1c)n, and when m_2 < cn, for some constant c > 0 not depending on anything.
 Publication:

arXiv eprints
 Pub Date:
 November 2008
 arXiv:
 arXiv:0811.4737
 Bibcode:
 2008arXiv0811.4737B
 Keywords:

 Mathematics  Combinatorics;
 11B50;
 11B75;
 05D05
 EPrint:
 27 pages, 3 figures