Steiner Triple Systems Intersecting in Pairwise Disjoint Blocks (original) (raw)
Related papers
The Fine Intersection Problem for Steiner Triple Systems
Graphs and Combinatorics, 2008
The intersection of two Steiner triple systems (X, A) and (X, B) is the set A ∩ B. The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m, n) such that there exist two Steiner triple systems of order v whose intersection I satisfies | ∪A∈I A| = m and |I| = n. We show that for v ≡ 1 or 3 (mod 6), |I(v)| = Θ(v 3), where previous results only imply that |I(v)| = Ω(v 2).
Sets of three pairwise orthogonal Steiner triple systems
Journal of Combinatorial Theory, Series A, 2003
Two Steiner triple systems (STS) are orthogonal if their sets of triples are disjoint, and two disjoint pairs of points defining intersecting triples in one system fail to do so in the other. In 1994, it was shown (Canad. J. Math. 46(2) (1994) 239-252) that there exist a pair of orthogonal Steiner triple systems of order v for all v 1; 3 (mod 6), with vX7; va9: In this paper we show that there exist three pairwise orthogonal Steiner triple systems of order v for all v 1 ðmod 6Þ; with vX19 and for all v 3 ðmod 6Þ; with vX27 with only 24 possible exceptions. r
Journal of Combinatorial Designs, 2005
In this paper, we present a conjecture that is a common generalization of the Doyen-Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u, v v 1, 3 ðmod 6Þ, u < v v < 2u þ 1, we ask for the minimum r such that there exists a Steiner triple system ðU, BÞ, jUj ¼ u such that some partial system ðU, Bnq qÞ can be completed to an STSðv vÞ, ðV, B 0 Þ, where jq qj ¼ r. In other words, in order to ''quasi-embed'' an STSðuÞ into an STSðv vÞ, we must remove r blocks from the small system, and this r is the least such with this property. One can also view the quantity ðuðu À 1Þ=6Þ À r as the maximum intersection of an STSðuÞ and an STSðv vÞ with u < v v. We conjecture that the necessary minimum r ¼ ðv v À uÞð2u þ 1 À v vÞ=6 can be achieved, except when u ¼ 6t þ 1 and v v ¼ 6t þ 3, in which case it is r ¼ 3t for t 6 ¼ 2, or r ¼ 7 when t ¼ 2. Using small examples and recursion, we solve the cases v v À u ¼ 2 and 4, asymptotically solve the cases v v À u ¼ 6, 8, and 10, and further show for given v v À u > 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v v À u). Some results are obtained for v v close to 2u þ 1 as well. The cases where v v % 3u=2 seem to be the hardest.
Independent sets in Steiner triple systems
Ars Combinatoria, 2004
A set of points in a Steiner triple system (STS(v)) is said to be independent if no three of these points occur in the same block. In this paper we derive for each k 8 a closed formula for the number of independent sets of cardinality k in an STS(v). We use the formula to prove that every STS(21) has
Tricyclic Steiner Triple Systems
Graphs and Combinatorics, 2010
A Steiner triple system of order v, denoted ST S(v), is said to be tricyclic if it admits an automorphism whose disjoint cyclic decomposition consists of three cycles. In this paper we give necessary and sufficient conditions for the existence of a tricyclic ST S(v) for several cases. We also pose conjectures concerning their existence in two remaining cases. Keywords Steiner triple systems • Tricyclic automorphism 1 Introduction A Steiner triple system of order v, denoted ST S(v), is a v-element set, X , of points, together with a set β, of unordered triples of elements of X , called blocks, such that any two points of X are together in exactly one block of β. It is well known that a ST S(v) exists if and only if v ≡ 1 or 3 (mod 6). For a general review of triple systems in general, see [5]. An automorphism of a ST S is a permutation π of X which fixes β. A permutation π of a v-element set is said to be of type [π ] = [π 1 , π 2 ,. .. , π v
On the Block Coloring of Steiner Triple Systems
Journal of Mathematical Extension, 2014
A Steiner triple system of order v, STS(v), is an ordered pair S = (V, B), where V is a set of size v and B is a collection of triples of V such that every pair of V is contained in exactly one triple of B. A k-block coloring is a partitioning of the set B into k color classes such that every two blocks in one color class do not intersect. In this paper, we introduce a construction and use it to show that for every k-block colorable STS(v) and l-block colorable STS(w), there exists a (k+lv)-block colorable STS(vw). Moreover, it is shown that for every kblock colorable STS(v), every STS(2v+1) obtained from the well-known construction is (k + v)-block colorable.