Geodesic
Construction of Steiner Triple Systems Having Exactly One Triple in Common
Canadian journal of mathematics, Tome 26 (1974) no. 1, pp. 225-232

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A Steiner triple system is a pair (Q, t) where Q is a set and t a collection of three element subsets of Q such that each pair of elements of Q belong to exactly one triple of t. The number |Q| is called the order of the Steiner triple system (Q, t). It is well-known that there is a Steiner triple system of order n if and only if n ≡ 1 or 3 (mod 6). Therefore in saying that a certain property concerning Steiner triple systems is true for all n it is understood that n ≡ 1 or 3 (mod 6). Two Steiner triple systems (Q, t 1) and (Q, t 2) are said to be disjoint provided that t 1 ∩ t 2 = Ø. Recently, Jean Doyen has shown the existence of a pair of disjoint Steiner triple systems of order n for every n ≧ 7 [1]. 
Lindner, Charles C. Construction of Steiner Triple Systems Having Exactly One Triple in Common. Canadian journal of mathematics, Tome 26 (1974) no. 1, pp. 225-232. doi: 10.4153/CJM-1974-022-9
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