Pasch trades with a negative block

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.

► We define equivalence of STS(v)s under Pasch trades with a negative block. ► Any 2 STS(v)s equivalent under cycle trades are proved equivalent in this sense. ► Any 2 isomorphic STS(v)s on the same base set are proved equivalent. ► Over 100 trades on up to 10 blocks can be effected in this way. ► There are trades which cannot be effected in this way.