tailieunhanh - Báo cáo toán học: "A New Method to Construct Lower Bounds for Van der Waerden Numbers"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A New Method to Construct Lower Bounds for Van der Waerden Numbers. | A New Method to Construct Lower Bounds for Van der Waerden Numbers . Herwig . Heule . van Lambalgen H. van Maaren Department of Electrical Engineering Mathematics and Computer Science Delft University of Technology The Netherlands marijn@ Submitted Nov 1 2005 Accepted Dec 18 2006 Published Jan 3 2007 Mathematics Subject Classification 05D10 Abstract We present the Cyclic Zipper Method a procedure to construct lower bounds for Van der Waerden numbers. Using this method we improved seven lower bounds. For natural numbers r k and n a Van der Waerden certificate W r k n is a partition of 1 . n into r subsets such that none of them contains an arithmetic progression of length k or larger . Van der Waerden showed that given r and k a smallest n exists - the Van der Waerden number W r k - for which no certificate W r k n exists. In this paper we investigate Van der Waerden certificates which have certain symmetrical and repetitive properties. Surprisingly it shows that many Van der Waerden certificates which must avoid repetitions in terms of arithmetic progressions reveal strong regularities with respect to several other criteria. The Cyclic Zipper Method exploits these regularities. To illustrate these regularities two techniques are introduced to visualize certificates. Supported by the Dutch Organization for Scientific Research NWO under grant THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R6 1 1 Introduction In 1927 the Dutch mathematician Van der Waerden proved 18 a generalization of a conjecture of Schur1 For given numbers r and k there exists a smallest number n - the Van der Waerden number W r k - such that each partition of the set 1 2 . ng into r subsets contains at least one subset with an arithmetic progression of at least length k. An arithmetic progression of length k is a sequence of k numbers such that the differences between consecutive numbers