Fragments in symmetric configurations with block size 3

TL;DR

This paper investigates the structure of symmetric configurations with block size 3, focusing on the occurrence of triangles, and establishes conditions for their existence and properties.

Contribution

It introduces formulas for counting block occurrences, identifies the triangle as a fundamental basis, and characterizes when symmetric configurations without triangles exist.

Findings

Triangles form a basis in these configurations.

Configurations without triangles exist only for specific sizes (v=15 or v≥17).

Analysis of configurations with cyclic automorphisms and their triangle content.

Abstract

We begin the study of collections of three blocks which can occur in a symmetric configuration with block size 3, $v_3$. Formulae are derived for the number of occurrences of these and it is shown that the triangle, i.e. abf, ace, bcd is a basis. It is also shown that symmetric configurations without triangles exist if and only if $v=15$ or $v \geq 17$. Such configurations containing "many" triangles are also discussed and a complete analysis of the triangle content of those with a cyclic automorphism is given.