Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Subsums of a Zero-sum Free Subset of an Abelian Group. | Subsums of a Zero-sum Free Subset of an Abelian Group Weidong Gao1 Yuanlin Li2 Jiangtao Peng3 and Fang Sun4 1 3 4Center for Combinatorics LPMC Nankai University Tianjin . China 2Department of Mathematics Brock University St. Catharines Ontario Canada L2S 3A1 1gao@ 2yli@ 3pjt821111@ 4sunfang2005@ Submitted Mar 22 2008 Accepted Sep 2 2008 Published Sep 15 2008 Mathematics Subject Classification 11B Abstract Let G be an additive finite abelian group and S c G a subset. Let f S denote the number of nonzero group elements which can be expressed as a sum of a nonempty subset of S. It is proved that if S 6 and there are no subsets of S with sum zero then f S 19. Obviously this lower bound is best possible and thus this result gives a positive answer to an open problem proposed by . Eggleton and P. Erdos in 1972. As a consequence we prove that any zero-sum free sequence S over a cyclic group G of length S op8 contains some element with multiplicity at least . 1 Introduction and Main Results Let G be an additive abelian group and S c G a subset. We denote by f G S f S the number of nonzero group elements which can be expressed as a sum of a nonempty subset of S. For a positive integer k 2 N let F k denote the minimum of all f A T where the minimum is taken over all finite abelian groups A and all zero-sum free subsets T c A with TI k. This invariant F k was first studied by . Eggleton and P. Erdos in 1972 see 4 . For every k 2 N they obtained a subset S in a cyclic group G with S I k such F k f G S j 1 k 1 a detailed proof may be found in 8 Section and . Olson 10 proved that F k 1 . 9 THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R116 1 Moreover Eggleton and Erdos determined F k for all k 5 and they stated the following conjecture which holds true for k 5 Conjecture . For every k 2 N there is a cyclic group G and a zero-sum free subset S G G with SI k such that F k f G S . Eggleton and Erdos .
