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The Fine Intersection Problem for Steiner Triple Systems

Authors:	Yeow Meng Chee Alan C H Ling Hao Shen

Institution:	(1) Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637616, Singapore;(2) Card View Pte. Ltd., 41 Science Park Road, #04-08A The Gemini, Singapore Science Park II, Singapore, 117610, Singapore;(3) Department of Computer Science, University of Vermont, Burlington, Vermont 05405, USA;(4) Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200030, People’s Republic of China

Abstract:	The intersection of two Steiner triple systems $(X,{\mathcal{A}})$ and $(X,{\mathcal{B}})$ is the set ${\mathcal{A}}\cap{\mathcal{B}}$ . The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m, n) such that there exist two Steiner triple systems of order v whose intersection ${\mathcal{I}}$ satisfies $\|\cup_{A\in{\mathcal{I}}} A\|=m$ and $\|{\mathcal{I}}\|=n$ . We show that for v ≡ 1 or 3 (mod 6), \|I(v)\| = Θ(v ³), where previous results only imply that \|I(v)\| = Ω(v ²). Received: January 23, 2006. Final Version received: September 2, 2006

Keywords:	Steiner triple systems Intersection
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