Pasch trades with a negative block

Abstract

A Steiner triple system of order $v$, STS($v$), may be called equivalent to another STS($v$) if one can be converted to the other by a sequence of three simple operations involving Pasch trades with a single negative block. It is conjectured that any two STS($v$)s on the same base set are equivalent in this sense. We prove that the equivalence class containing a given system $S$ on a base set $V$ contains all the systems that can be obtained from $S$ by any sequence of well over one hundred distinct trades, and that this equivalence class contains all isomorphic copies of $S$ on $V$. We also show that there are trades which cannot be effected by means of Pasch trades with a single negative block.