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Minimal homogeneous Steiner 2-(v,3) trades

Authors:	Nicholas J Cavenagh Diane M Donovan Emine Şule Yazıcı

Institution:	1. School of Mathematics, The University of New South Wales, NSW 2052, Australia;2. Centre for Discrete Mathematics and Computing, University of Queensland, St Lucia 4072, Australia;3. Koç University Rumeli Feneri Yolu, 34450 Sar?yer Istanbul, Turkey;1. UK;2. UK;3. UK;4. UK;5. UK;6. UK

Abstract:	A Steiner 2- $(v, 3)$ trade is a pair $(T_{1}, T_{2})$ of disjoint partial Steiner triple systems, each on the same set of $v$ points, such that each pair of points occurs in $T_{1}$ if and only if it occurs in $T_{2}$ . A Steiner 2- $(v, 3)$ trade is called d-homogeneous if each point occurs in exactly d blocks of $T_{1}$ (or $T_{2}$ ). In this paper we construct minimal d-homogeneous Steiner 2- $(v, 3)$ trades of foundation $v$ and volume $dv / 3$ for sufficiently large values of $v$ . (Specifically, $v > 3 (1.75 d^{2} + 3)$ if $v$ is divisible by 3 and $v > d (4^{d / 3 + 1} + 1)$ otherwise.)

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