5-sparse Steiner Triple Systems of Order n Exist for Almost All Admissible n

Steiner triple systems are known to exist for orders n ≡ 1, 3 mod 6, the admissible orders. There are many known constructions for infinite classes of Steiner triple systems. However, Steiner triple systems that lack prescribed configurations are harder to find. This paper gives a proof that the spectrum of orders of 5-sparse Steiner triple systems has arithmetic density 1 as compared to the admissible orders. 1 Background Let v ∈ N and let V be a v-set. A Steiner triple system of order v, abbreviated STS(v), is a collection B of 3-sets of V , called blocks or triples, such that every distinct pair of elements of V lies in exactly one triple of B. An STS(v) exists exactly when v ≡ 1 or 3 mod 6, the admissible orders. Wilson [13] showed that the number of non-isomorphic Steiner triple systems of order n is asymptotically at least (e−5n)n 2/6. Much less is known about the existence of Steiner triple systems that avoid certain configurations. An rconfiguration of a system is a set of r distinct triples whose union consists of no more than r + 2 points. A Steiner triple system that lacks r-configurations is said to be r-sparse. In other words, a Steiner triple system where the union of every r distinct triples has at least r + 3 points is r-sparse. In 1976, Paul Erdős conjectured that for any r > 1, there exists a constant Nr such that whenever v > Nr and v is an admissible order, an r-sparse STS(v) exists[4]. The statement is trivial for r = 2, 3. For r = 4, there is only one type of 4-configuration, a Pasch. Paschs have the form: {a, b, c}, {a, d, e}, {f, b, d}, {f, c, e} (1) ∗Thanks to the editors of this journal for considering this for publication. the electronic journal of combinatorics 12 (2005), #R68 1 In this paper, Paschs are written in the order presented above. Viewing a Steiner triple system as a 3-regular hypergraph with the point-set of the graph being the points of the Steiner triple system and the edge-set being the triples, we can graphically represent the system by plotting the point set as vertices and connecting the three vertices of an edge (triple) by a smooth line. With this in mind, a Pasch as in (1) can be graphically represented as: