Minimal homogeneous Steiner 2-(upsilon,3) trades

Abstract

A Steiner 2-(upsilon, 3) trade is a pair (T-1, T-2) of disjoint partial Steiner triple systems, each on the same set of upsilon points, such that each pair of points occurs in T-1 if and only if it occurs in T-2. A Steiner 2-(upsilon, 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-(upsilon, 3) trades of foundation upsilon and volume d upsilon/3 for sufficiently large values of upsilon. (Specifically, upsilon > 3(1.75d(2) + 3) if upsilon is divisible by 3 and upsilon > d(4(d/3+1) + 1) otherwise.) (C) 2007 Elsevier B.V. All rights reserved.