Generalized pentagonal geometries

TL;DR

This paper introduces a generalized framework for pentagonal geometries by integrating Steiner systems, expanding the classical concept to include more complex point-line incidence structures.

Contribution

It extends pentagonal geometries to include Steiner systems, broadening the scope of partial linear spaces with new combinatorial configurations.

Findings

Defined generalized pentagonal geometries with Steiner systems

Established conditions for the existence of these geometries

Provided examples illustrating the new structures

Abstract

A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. Here we generalize the concept by allowing the points not collinear with $x$ to form the point set of a Steiner system $S(2,k,w)$ whose blocks are lines of the geometry.