Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On zero-sum free subsets of length 7. | On zero-sum free subsets of length 7 Pingzhi Yuan School of Mathematics South China Normal University Guangzhou 510631 mcsypz@ Xiangneng Zeng Department of Mathematics Sun Yat-Sen University Guangzhou 510275 Submitted Nov 2 2009 Accepted Jul 26 2010 Published Aug 9 2010 Mathematics Subject Classifications primary 11B75 secondary 11B50 Abstract Let G be a finite additively written abelian group and let X be a subset of 7 elements in G. We show that if X contains no nonempty subset with sum zero then the number of the elements which can be expressed as the sum over a nonempty subsequence of X is at least 24. 1 Introduction Let G be an additive abelian group and X c G a subset of G. We denote by f G X f X the number of nonzero group elements which can be expressed as a sum of a nonempty subset of X. For a positive integer k G N let f k denote the minimum of all f G X where the minimum is taken over all finite abelian groups G and all zero-sum free subsets X c G with X k. The invariant f k was first studied by R. B. Eggleton and P. Erdos in 1972 1 . For every k G N they obtained a subset X in a cyclic group G with X k such that f k f G X 1 k2 1. 1 And J. E. Olson 2 proved that f k 9k2. Moreover Eggleton and Erdos determined f k for all k 5 and they stated the following conjecture which holds true for k 5 Supported by NSF of China No. 10971072 and by the Guangdong Provincial Natural Science Foundation No. 8151027501000114 . THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R104 1 Conjecture . For every k E N there is a cyclic group G and a zero-sum free subset X c G with X k such that f k f G X . Recently Weidong Gao et al. 3 proved that f 6 19 and et al. 5 showed that f G X 24 the lower bound is sharp where G is a cyclic group X 7. Together with the conjecture above we have that f 7 24. The main aim of the present paper is to show the following theorem. Theorem . f 7 24. In Section 2 we fix the notation. Sections 3 and
