tailieunhanh - Báo cáo toán học: "A van der Waerden Variant"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A van der Waerden Variant. | A van der Waerden Variant Kevin J. Compton BRICS Research Centre University of Aarhus Denmark and EECS Department University of Michigan Ann Arbor MI 48109-2122 kjc@ Abstract The classical van der Waerden Theorem says that for every every finite set S of natural numbers and every fc-coloring of the natural numbers there is a monochromatic set of the form aS b for some a 0 and b 0. . monochromatism is obtained by a dilation followed by a translation. We investigate the effect of reversing the order of dilation and translation. S has the variant van der Waerden property for fc colors if for every fc-coloring there is a monochromatic set of the form a S b for some a 0 and b 0. On the positive side it is shown that every two-element set has the variant van der Waerden property for every fc. Also for every finite S and fc there is an n such that nS has the variant van der Waerden property for fc colors. This extends the classical van der Waerden Theorem. On the negative side it is shown that if S has at least three elements the variant van der Waerden property fails for a sufficiently large fc. The counterexamples to the variant van der Waerden property are constructed by specifying colorings as Thue-Morse sequences. Submitted July 17 1997 Accepted April 2 1999. AMS Subject Classification. Primary 05D10. Secondary 11B85 68R15. 1 Introduction. Van der Waerden s theorem on arithmetic progressions is over seventy years old 26 but it continues to reveal new facets and inspire new results. It has many generalizations such as the Hales-Jewett Theorem 6 and multidimensional versions 21 . It has had unexpected connections with other parts of mathematics such as topological dynamics 5 . The numerical bounds from van der Waerden s original proof long thought to be the best attainable have been dramatically reduced in recent years 23 . . In its most familiar formulation van der Waerden s Theorem says that if N 0 1 2 . is partitioned into a finite number of classes one