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### Steiner Triple System

Let be a set of elements together with a set of 3-subset (triples) of such that every 2-subset of occurs in exactly one triple of . Then is called a Steiner triple system and is a special case of a Steiner system with and . A Steiner triple system of order exists iff (Kirkman 1847). In addition, if Steiner triple systems and of orders and exist, then so does a Steiner triple system of order (Ryser 1963, p. 101).Examples of Steiner triple systems of small orders are(1)(2)(3)The Steiner triple system is illustrated above.The numbers of nonisomorphic Steiner triple systems of orders , 9, 13, 15, 19, ... (i.e., ) are 1, 1, 2, 80, 11084874829, ... (Stinson and Ferch 1985; Colbourn and Dinitz 1996, pp. 14-15; Kaski and Östergård 2004; OEIS A030129). is the same as the finite projective plane of order 2. is a finite affine plane which can be constructed from the array(4)One of the two s is a finite hyperbolic plane. The 80 Steiner triple systems..

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