Pasch trades with a negative block

Drizen, A.L.; Grannell, Mike and Griggs, Terry (2011). Pasch trades with a negative block. Discrete Mathematics, 311(21) pp. 2411–2416.

DOI: https://doi.org/10.1016/j.disc.2011.06.023

Abstract

A Steiner triple system of order $v$ , STS( $v$ ), may be called $\emph{equivalent}$ 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.