On a smallest topological triangle free (n_4) point-line configuration

Jürgen Bokowski, Hendrik Van Maldeghem
2020 The Art of Discrete and Applied Mathematics  
We study an abstract object, a finite generalised quadrangle W (3), due to Jacques Tits, that can be seen as the Levi graph of a triangle free (40 4 ) point-line configuration. We provide for W (3) representations as a topological (40 4 ) configuration, as a (40 4 ) circle representation, and a representation in the complex plane. These come close to a still questionable (real) geometric (40 4 ) point-line configuration realising this finite generalised quadrangle. This abstract (40 4 )
more » ... ation has interesting triangle free realisable geometric subconfigurations, which we also describe. A topological (n 4 ) configuration for n < 40 must contain a triangle, so our triangle free example is minimal. A c c e p t e d m a n u s c r i p t 2 Art Discrete Appl. Math. Definition 1.1. An (n k ) configuration is a set of n points and n lines such that every point lies on precisely k of these lines and every line contains precisely k of these points. We distinguish three concepts. Definition 1.2. When the lines are straight lines in the projective plane, we have a geometric (n k ) configuration. Definition 1.3. When the lines are pseudolines forming a rank 3 oriented matroid, we have a topological (n k ) configuration. Definition 1.4. When the lines are abstract lines, we have an abstract (n k ) configuration. We assume the reader to know basic facts about rank 3 oriented matroids or pseudoline arrangements in the real projective plane. This article provides, among other results, a triangle free topological (40 4 ) configuration. We remark that triangle free configurations have been studied so far only for smaller (n 3 ) configurations, see e.g. [3], [10], or [15]. Definition 1.5. The generalised quadrangle W (3) is the point-line geometry where the points are the points of the projective 3-space P 3 (3) over the field of 3 elements, and the lines are the lines of P 3 (3) fixed under a symplectic polarity. A symplectic polarity is a permutation of the set of points, lines and planes of P 3 (3) mapping the points to planes, lines to lines and planes to points, such that incidence and non-incidence are both preserved (that is, containment of points in lines and planes, and of lines in planes is transferred into reversed containment), and the permutation has order 2, that is, if a point p is mapped to the plane α, then the plane α is mapped to the point p. Such a polarity can be described, after suitable coordinatization, as mapping the point (a, b, c, d) to the plane with equation bX − aY + dZ − cU = 0, from which all other images follow.
doi:10.26493/2590-9770.1355.f3d fatcat:efpioe5wszgvfpeso4muebko2q