On the number of small Steiner triple systems with Veblen points

TL;DR

This paper investigates the quantity of Steiner triple systems with Veblen points for specific orders, providing counts and examples, advancing understanding of their structure and enumeration.

Contribution

It introduces methods to count and construct Steiner triple systems with Veblen points for orders 19, 27, and 31, addressing an open enumeration problem.

Findings

Number of such systems for order 19, 27, 31

Examples of Steiner triple systems with Veblen points

Insights into their structural properties

Abstract

The concept of Schreier extensions of loops was introduced in the general case in [11] and, more recently, it has been explored in the context of Steiner loops in [6]. In the latter case, it gives a powerful method for constructing Steiner triple systems containing Veblen points. Counting all Steiner triple systems of order v is an open problem for v>21. In this paper, we investigate the number of Steiner triple systems of order 19, 27 and 31 containing Veblen points and we present some examples.