The embedding problem for partial Steiner triple systems

The system has the nice property that any pair of distinct elements of V occurs in exactly one of the subsets. This makes it an example of a Steiner triple system. Steiner triple systems first appeared in the mathematical literature in the mid-nineteenth century but the concept must surely have been thought of long before then. An excellent historical introduction appears in [7]. As pointed out there, it is interesting that “triple systems find their origins in studies of cubic curves, rather than in recreational problems as is often thought”. A Steiner triple system is more formally defined as a pair (V,B) where V is a finite set and B is a set of 3-element subsets of V such that each 2-element subset of V is a subset of exactly one of the 3-element subsets in B. The elements of B are called triples and |V | is the order of the system. If there is a Steiner triple system of order v then simple counting establishes that it contains v(v − 1)/6 triples, and each element of v occurs in (v − 1)/2 triples. It follows that if there is a Steiner triple system of order v, then v ≡ 1 or 3 (mod 6). Such integers are called admissible. In 1847 Kirkman [11] proved the existence of Steiner triple systems of all admissible orders. Steiner triple systems of orders 1 and 3 are trivial. Up to isomorphism, the number N(v) of Steiner triple systems of order v for v = 7, 9, 13, 15, 19 is given in the following table, see [7, 10]. For v > 19 the exact value of N(v) is unknown.