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
have been studied by Tonchev and Weishaar (1997).
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Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, pp. 14-15
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2075-2092, 2004.Kaski, P.; Östergård, P. R. J.;
Topalova, S.; and Zlatarski, R. "Steiner Triple Systems of Order 19 and 21 with
Subsystems of Order 7." http://www.tcs.hut.fi/~pkaski/19and21.ps.Kirkman,
T. P. "On a Problem in Combinatorics." Cambridge Dublin Math. J.2,
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On-Line Encyclopedia of Integer Sequences."Stinson, D. R.
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