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, v3. Formulae are derived for the number of their occurrences 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>=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.