Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Monochromatic and zero-sum sets of nondecreasing modified diameter. | Monochromatic and zero-sum sets of nondecreasing modified diameter David Grynkiewicz and Rasheed Sabah Ỉ Submitted Oct 24 2004 Accepted Mar 24 2006 Published Mar 30 2006 Mathematics Subject Classification 05D05 11B75 Abstract Let m be a positive integer whose smallest prime divisor is denoted by p and let Zm denote the cyclic group of residues modulo m. For a set B x1 x2 xm of m integers satisfying x1 x2 xm and an integer j satisfying 2 j m define gj B xj x1. Furthermore define fj m 2 define fj m Zm to be the least integer N such that for every coloring A 1 N 0 1 every coloring A 1 N Zm there exist two m-sets B1 B2 c 1 N satisfying i max B1 min B2 ii gj B1 gj B2 and iii A Bj 1 for i 1 2 and iii xeB A x 0 for i 1 2 . We prove that fj m 2 5m 3 for all j with equality holding for j m and that fj m Zm 8m m 6. Moreover we show that fj m 2 4m 2 j 1 k where k 1 1 A 8m 9 M 2 I and if m j 1 is prime or j m p 1 that fj m Zm 6m 4. We conclude by showing fm-i m 2 fm-i m Zm for m 9. 1 Introduction Let a b denote the set of integers between a and b inclusive. For a set S an S-coloring of 1 A is a function A 1 A S. If S 0 1 . r 1 then we call A an r-coloring. The following is the Erdos-Ginzburg-Ziv EGZ theorem 1 14 30 . Theorem 0. Let m be a positive integer. If A 1 2m 1 Zm then there exist m distinct integers x1 x2 . xm 2 1 2m 1 such that 2 A xi 0. Moreover 2m 1 is the smallest number for which the above assertion holds. Department of Mathematics Caltech Pasadena CA 91125 Department of Mathematics Harvard University Cambridge MA 02138 iThe second author was funded by NSF grant DMS0097317. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R28 1 The EGZ theorem can be viewed as a generalization of the pigeonhole principle for 2 boxes since the m-term zero-sum subsequences of a sequence consisting only of 0 s and 1 s are exactly the monochromatic m-term subsequences . As such several theorems of Ramsey-type have been generalized similarly by considering Zm-colorings and zerosum .
