Distance and fractional isomorphism in Steiner triple systems

A.D. Forbes, M.J. Grannell and T.S. Griggs Department of Mathematics The Open University Walton Hall, Milton Keynes MK7 6AA UNITED KINGDOM [email protected] [email protected] [email protected] Abstract In [8], 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 v0(N) such that for all admissible v ≥ v0(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 ≥ 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.