Abstract
In [8], Quattrochi and Rinaldi introduced the idea ofn −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 integerN, there existsv 0(N) such that for all admissiblev≥v 0(N) and for each STS(v) (sayS), there exists an STS(v) (sayS′) such that for somen>N, S is strictlyn −1-isomorphic toS′. We also prove that for all admissiblev≥13, there exist two STS(v)s which are strictly 2−1-isomorphic.
Define the distance between two Steiner triple systemsS andS′ of the same order to be the minimum volume of a tradeT which transformsS into a system isomorphic toS′. 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.
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Forbes, A.D., Grannell, M.J. & Griggs, T.S. Distance and fractional isomorphism in Steiner triple systems. Rend. Circ. Mat. Palermo 56, 17–32 (2007). https://doi.org/10.1007/BF03031425
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DOI: https://doi.org/10.1007/BF03031425