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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Maximum subsets of (0, 1] with no solutions to x + y = kz. | Maximum subsets of 0 1 with no solutions to x y kz Fan R. K. Chung Department of Mathematics University of Pennsylvania Philadelphia PA 19104 John L. Goldwasser West Virginia University Morgantown WV 26506 Abstract If k is a positive real number we say that a set S of real numbers is k-sum-free if there do not exist X y z in S such that X y kz. For k greater than or equal to 4 we find the essentially unique measurable k-sum-free subset of 0 1 of maximum size. 1 Introduction We say that a set S of real numbers is sum-free if there do not exist X y z is S such that X y z. If k is a positive real number we say that a set S of real numbers is k-sum-free if there do not exist X y z in S such that X y kz we require that not all X y and z be equal to each other to avoid a meaningless problem when k 2 . Let f n k denote the maximum size of a k-sum-free subset of 1 2 . ng. It is easy to show 1 2 that f n 1 n . For k 1 and n odd there are precisely two such maximum sets the odd integers and the top half. For n even and greater than 9 there are precisely three such sets see 1 the two maximum sets for the odd number n 1 and the top half. The problem of determining f n 2 is unsolved. Roth 4 proved that a subset of the positive integers with positive upper density contains three-term arithmetic progressions. The current best bounds for f n 2 were established by Salem and Spencer 5 and Heath-Brown and Szemeredi 3 . 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 3 1996 R1 2 Chung and Goldwasser 1 proved a conjecture of Erdos that f n 3 is n 2 roughly n. They showed that f n 3 the set of odd integers less than or equal to n is the unique maximum set. for n 4 and that for n 23 Loosely speaking the set of odd numbers less than or equal to n qualihes as a k-sum-free set for odd k because of parity considerations while the top half maximum sum-free set qualihes because of magnitude considerations the sum of two numbers in the top half is too big. There is an obvious way to take a magnitude
