A complete classification of the $(15_4 20_3)$-configurations with at least three $K_5$-graphs

TL;DR

This paper classifies all configurations with 15 points containing at least three K5 graphs, linking them to systems of triangle perspectives and providing a complete enumeration and automorphism analysis.

Contribution

It provides a complete classification of (15_4 20_3)-configurations with at least three K5 graphs, including automorphism groups.

Findings

Classified all such configurations on 15 points.

Identified configurations as systems of triangle perspectives.

Determined automorphism groups of these configurations.

Abstract

The class of $\left(\binom{n+1}{2}_{n-1} \binom{n+1}{3}_3\right)$-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.