Distance and fractional isomorphism in Steiner triple systems

Forbes, Anthony; Grannell, Mike and Griggs, Terry (2007). Distance and fractional isomorphism in Steiner triple systems. Rendiconti del Circolo Matematico di Palermo Serie II, 56, pp. 17–32.

Abstract

Quattrochi and Rinaldi introduced the idea of $n^{-1}$ - isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. The main result is that for any positive integer $N$ , there exists $v_0(N)$ such that for all admissible $v \ge v_0(N)$ and for each STS $(v)$ (say $S$ ), there exists an STS $(v)$ (say $S'$ ) such that for some $n > N$ , $S$ is strictly $n^{-1}$ -isomorphic to $S'$ . We also prove that for all admissible $v \ge 13$ , there exist two STS $(v)$ s which are strictly $2^{-1}$ -isomorphic. Define the distance between two Steiner triple systems $S$ and $S'$ of the same order to be the minimum volume of a trade $T$ which transforms $S$ into a system isomorphic to $S'$ . We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly $2^{-1}$ -isomorphic and $3^{-1}$ -isomorphic pairs of STS $(15)$ s.