Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The universal embedding of the near polygon Gn. | The universal embedding of the near polygon Gn Bart De Bruyn Department of Pure Mathematics and Computer Algebra Ghent University Gent Belgium bdb@ Submitted Oct 9 2006 Accepted May 16 2007 Published May 23 2007 Mathematics Subject Classihcations 05B25 51A45 51A50 51E12 Abstract In 12 it was shown that the dual polar space DH 2n 1 4 n 2 has a sub near-2n-gon Gn with a large automorphism group. In this paper we determine the absolutely universal embedding of this near polygon. We show that the generating and embedding ranks of Gn are equal to 2n. We also show that the absolutely universal embedding of Gn is the unique full polarized embedding of this near polygon. 1 Introduction Definitions A near polygon is a partial linear space S P L I I c P X L with the property that for every point p and every line L there exists a unique point on L nearest to p. Here distances are measured in the point or collinearity graph r of S. If d is the diameter of r then the near polygon is called a near 2d-gon. A near 0-gon is just a point and a near 2-gon is a line. Near quadrangles are usually called generalized quadrangles Payne and Thas 23 . Near polygons were introduced by Shult and Yanushka in 26 . For a discussion on the basic theory of near polygons we refer to the recent book 13 of the author. A near polygon is called dense if every line is incident with at least three points and if every two points at distance 2 from each other have at least two common neighbours. By Theorem 4 of Brouwer and Wilbrink 6 every two points of a dense near 2d-gon at distance Ỗ 2 0 . dg from each other are contained in a unique convex sub-2Vgon. These convex sub-2Vgons are called quads if Ỗ 2 hexes if Ỗ 3 and maxes if Ỗ d 1. Postdoctoral Fellow of the Research Foundation - Flanders Belgium THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R39 1 The existence of quads in a dense near polygon was already shown by Shult and Yanushka 26 A proper convex subspace F of a dense near polygon
