Fragments in symmetric configurations with block size 3

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.