Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: A Ramsey Treatment of Symmetry. | A Ramsey Treatment of Symmetry T. Banakh O. Verbitsky t Ya. Vorobets Department of Mechanics and Mathematics Lviv University 79000 Lviv Ukraine E-mail tbanakh@ Submitted November 8 1999 Accepted August 15 2000. But seldom is asymmetry merely the absence of symmetry. Hermann Weyl Symmetry Abstract Given a space Q endowed with symmetry we define ms Q r to be the maximum of m such that for any r-coloring of Q there exists a monochromatic symmetric set of size at least m. We consider a wide range of spaces Q including the discrete and continuous segments 1 . ng and 0 1 with central symmetry geometric figures with the usual symmetries of Euclidean space and Abelian groups with a natural notion of central symmetry. We observe that ms 1 . ng r and ms 0 1 r are closely related prove lower and upper bounds for ms 0 1 2 and find asymptotics of ms 0 1 r for r increasing. The exact value of ms Q r is determined for figures of revolution regular polygons and multi-dimensional parallelopipeds. We also discuss problems of a slightly different flavor and in particular prove that the minimal r such that there exists an r-coloring of the k-dimensional integer grid without infinite monochromatic symmetric subsets is k 1. MR Subject Number 05D10 Research supported in part by grant INTAS-96-0753. tPart of this work was done while visiting the Institute of Information Systems Vienna University of Technology supported by a Lise Meitner Fellowship of the Austrian Science Foundation FWF . 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R52 2 0 Introduction The aim of this work is given a space with symmetry to compute or to estimate the maximum size of a monochromatic symmetric set that exists for any r-coloring of the space. More precisely let Q be a space with measure ụ. Suppose that Q is endowed with a family 5 of transformations s Q Q called symmetries. A set B c Q is symmetric if s B B for a symmetry s 2 5. An r-coloring of Q is a map X Q 1 2 . r where each color class
