tailieunhanh - Báo cáo toán học: "The Structure of Maximum Subsets of {1, . . . , n} with No Solutions to a + b = kc"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: The Structure of Maximum Subsets of {1, . . . , n} with No Solutions to a + b = kc. | The Structure of Maximum Subsets of 1 . ng with No Solutions to a b kc Andreas Baltz Mathematisches Seminar University of Kiel D-24098 Kiel Germany aba@ Peter Hegarty Jonas Knape Urban Larsson Department of Mathematics Chalmers University of Technology Goteborg Sweden hegarty@ md9jonas md0larurg@ Tomasz Schoen Wydzial Matematyki i Informatyki Adam Mickiewicz University Poznan Poland schoen@ Submitted Nov 9 2004 Accepted Apr 22 2005 Published Apr 28 2005 MR Subject Classifications 05D05 11P99 Abstract If k is a positive integer we say that a set A of positive integers is k-sum-free if there do not exist a b c in A such that a b kc. In particular we give a precise characterization of the structure of maximum sized k-sum-free sets in 1 . ng for k 4 and n large. 1 Introduction A set of positive integers is called k-sum-free if it does not contain elements a b c such that a b kc supported by DFG Grant SR 7 9-2 research partially supported by KBN Grant 2 PO3A 007 24 THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R19 1 where k is a positive integer. Denote by f n k the maximum cardinality of a k-sum-free set in . ng. For k 1 these extremal sets are well-known Deshoulliers Freiman Sós and Temkin 1 proved in particular that the maximum 1-sum-free sets in 1 . ng are precisely the set of odd numbers and the top half H . ng. For n 8 even n . n 1g forms the only additional extremal set. The famous theorem of Roth 4 gives f n 2 o n . Chung and Goldwasser 2 solved the case k 3 by showing that the set of odd integers is the unique extremal set for n 22. For k 4 they gave an example of a k-sum-free set 3 of cardinality k k 2n fc fc2_28 fe4- 2fe2-4 n ỡ 1 which implies limn. V kkk-2 k k2_l i_-2k2_4 and they conjectured that this lower bound is the actual value. Moreover they conjectured that extremal k-sum-free sets consist of three intervals of consecutive integers with slight modihcations at the end-points if n is .