Krzysztof Petelczyc

Twisted Fano spaces and their classification, linear completions of systems of triangle perspectives

Twisted Fano spaces i.e. linear spaces with the parameters of PG(3, 2) which contain a pencil of Fano subplanes are completely classified and characterized. In particular, it is proved that twisted Fano spaces are exactly all the linear completions of systems of triangle perspectives with point degree 4.

103-configurations and projective realizability of multiplied configurations

Some remarks on 103-configurations which contain the complete graph K4 are given, on their representations, and on projective realizability. Results are applied to show a class of configurations that cannot be realized in any Pappian projective space.

A complete classification of the ( 15 4 20 3 ) -configurations with at least three K 5 -graphs

The class of ( ( n + 1 2 ) n - 1 ( n + 1 3 ) 3 ) -configurations which contain at least n - 2 K n -graphs coincides with the class of so called systems of triangle perspectives i.e.?of configurations which contain a bundle of n - 2 Pasch configurations with a common line. For n = 5 the class consists of all binomial partial Steiner triple systems on 15 points, that contain at least three K 5 -graphs. In this case a complete classification of respective configurations is given and their automorphisms are determined.

The complement of a point subset in a projective space and a Grassmann space

Abstract In a projective space we fix some set of points, a horizon, and investigate the complement of that horizon. We prove, under some assumptions on the size of lines, that the ambient projective space, together with its horizon, both can be recovered in that complement. Then we apply this result to show something similar for Grassmann spaces.

On some generalization of the Möbius configuration

The Mobius (8 4 ) configuration is generalized in a purely combinatorial approach. We consider (2 n n ) configurations M ( n ,  φ ) depending on a permutation φ in the symmetric group S n . Classes of non-isomorphic configurations of this type are determined. The parametric characterization of M ( n ,  φ ) is given. The uniqueness of the decomposition of M ( n ,  φ ) into two mutually inscribed n -simplices is discussed. The automorphisms of M ( n ,  φ ) are characterized for n  ≥ 3 .

Tresses of polygons

Introduction The main task of our paper is to present some class of regular configurations, which can be represented as a result of splitting (in some sense) polygons. These structures appear, in fact, in the framework of a more general theory but here we shall not present this theory in many details and only sketch some principal ideas that lead to the definition of configurations being the topic of this paper. We believe that these configurations can be of some interest on their own. Some interesting classical configurations (see e.g. [3], [6], [9]) can be represented as self-inscribed polygons, or as families of (possibly cyclically) inscribed polygons (comp. e.g. [6]). An easy idea of producing drawing such configurations (demonstrative algorithms of construction) usually is best formalized in the language of cyclic groups. Simply because a regular kgon via its rotations can be identified with the group GV Then the technique of difference sets or, more generally, of quasi difference sets (cf. [11], [12]) can be applied. Clearly, the simplest quasi difference set in a group Ck consists of 0 and 1 this one determines the fc-gon (cf. [7]). Similarly, simple quasi difference sets in products Ckx