Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers. | Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers Harold Fredricksen Naval Postgraduate School Department of Mathematics Monterey CA 93943 USA halF@ Melvin M. Sweet Institute for Defense Analyses Center for Communications Research 4320 Westerra Court San Diego CA 92121 USA sweet@ Submitted May 9 2000 Accepted May 18 2000 Abstract We give new lower bounds for the Schur numbers S 6 and S 7 . This will imply new lower bounds for the Multicolor Ramsey Numbers R6 3 and R7 3 . We also make several observations concerning symmetric sum-free partitions into 5 sets. Introduction A set of integers is said to be sum-free if for all i and j i in the set the sum i j is not in the set. The Schur number S k is defined to be the maximum integer n for which the interval 1 n can be partitioned into k sum-free sets. 1 I. Schur 7 proved that S k is finite and that S k 3S k - 1 1 . In the framework of Ramsey theory Schur s proof yields the inequality S k Rk 3 - 2 1 2 where Rk n denotes the k-color Ramsey number defined to be the smallest integer such that any k-color edge coloring of the complete graph on Rk n vertices has at least one complete subgraph all of whose edges have the same color. 1Some authors define S k to be the smallest n for which there are no sum-free partitions into k sets. 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 7 2000 R32 2 It is easily verified that S 1 1 S 2 4 and S 3 13 and . Baumert 1 showed in 1961 with the aid of a computer that S 4 44. The best known bounds for S 5 are 160 S 5 315 3 The first inequality in 3 is due to G. Exoo 4 and the second follows from 2 and the bounds for R5 3 given in S. Radziszowski s survey paper 6 . For earlier work on lower bounds for S k see H. Fredricksen 5 and A. Beutelspacher and W. Brestovansky 2 . The best previously known lower bound S 6 481 for S 6 follows from 1 . At the end of this note we list constructions that show S 6 536 S 7 1680. Using 2 we obtain the following lower bounds
