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).
Ball, W. W. R. and Coxeter, H. S. M. Mathematical
Recreations and Essays, 13th ed. New York: Dover, pp. 107-109 and 274,
1987.Colbourn, C. J. and Dinitz, J. H. (Eds.). "Steiner
Triple Systems." §4.5 in CRC
Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, pp. 14-15
and 70, 1996.Gardner, M. "Mathematical Games: On the Remarkable
Császár Polyhedron and Its Applications in Problem Solving." Sci.
Amer.232, 102-107, May 1975.Kaski, P. and Östergård,
P. R. J. "The Steiner Triple Systems of Order 19." Math. Comput.73,
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,
191-204, 1847.Lindner, C. C. and Rodger, C. A. Design
Theory. Boca Raton, FL: CRC Press, 1997.Ryser, H. J. Combinatorial
Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 99-102, 1963.Sloane,
N. J. A. Sequence A030129 in "The
On-Line Encyclopedia of Integer Sequences."Stinson, D. R.
and Ferch, H. "2000000 Steiner Triple Systems of Order 19." Math. Comput.44,
533-535, 1985.Tonchev, V. D. and Weishaar, R. S. "Steiner
Triple Systems of Order 15 and Their Codes." J. Stat. Plan. Inference58,
207-216, 1997.
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).
of small orders 
are
is illustrated above.
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
s
is a finite hyperbolic plane. The 80 Steiner
triple systems 
have been studied by Tonchev and Weishaar (1997).
