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Báo cáo toán học: "Extremal subsets of {1, ..., n} avoiding solutions to linear equations in three variables"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Extremal subsets of {1, ., n} avoiding solutions to linear equations in three variables. | Extremal subsets of 1 . ng avoiding solutions to linear equations in three variables Peter Hegarty Chalmers University of Technology and Gothenburg University Gothenburg Sweden hegarty@math.chalmers.se Submitted Jul 9 2007 Accepted Oct 30 2007 Published Nov 5 2007 Mathematics Subject Classification 05D05 11P99 11B75 Abstract We refine previous results to provide examples and in some cases precise classifications of extremal subsets of 1 . n containing no solutions to a wide class of non-invariant homogeneous linear equations in three variables i.e. equations of the form ax by cz with a b c. 1 Introduction A well-known problem in combinatorial number theory is that of locating extremal subsets of 1 . n which contain no non-trivial solutions to a given linear equation L a1 X1 akxk b 1 where a1 . ak b 2 Z and their GCD is one. Most of the best-known work concerns just three individual homogeneous equations L1 X1 X2 2x3 L2 X1 X2 X3 X4 L3 X1 X2 X3 where the corresponding subsets are referred to respectively as sets without arithmetic progressions Sidon sets and sum-free sets. The idea to consider arbitrary linear equations L was first enunciated explicitly in a pair of articles by Ruzsa in the mid-1990s 10 11 . The only earlier reference of note would appear to be a paper of Lucht 6 concerning homogeneous equations in three variables though Lucht s article was only concerned with subsets of N. Following Ruzsa denote by rL n the maximum size of a subset of 1 . n which contains no non-trivial solutions to a given equation L. Let us pause here THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R74 1 to recall explicitly what we mean by a trivial solution to 1 the definition is also given in 10 . Such solutions can only arise when L is translation-invariant i.e. Xai b 0. Then a solution x1 . xk to 1 is said to be trivial if there is a partition of the index set 1 . k T t t Ti such that Xi Xj whenever i and j are in the same part of the partition and for each r 1 . l one has .