tailieunhanh - Báo cáo toán học: "On zero-sum free subsets of length 7"

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