Steiner Triple Systems Having a Prescribed Number of Triples in Common

Abstract. 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 ).

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“…We Intersection problems for STS remain well alive three decades after the seminal paper of Lindner and Rosa [6],…”

Section: Resultsmentioning

confidence: 99%

The Fine Intersection Problem for Steiner Triple Systems

Chee

Ling

Shen

2008

Graphs and Combinatorics

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“…It consists of seven lines and seven points, with three points on a line and, dually, three lines per point, where every pair of points is connected by a line, every line intersects every other line, and there are four points such that no line contains more than two of them. It is well known [1][2][3] that there are thirty different Fano planes on a given seven-element set. Slightly rephrased, there are thirty different ways to label the points of the Fano plane by integers from 1 to 7, two labeled Fano planes having zero, one or three lines in common, and each line occurring in six Fano planes.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Intricacies of Labeled Fano Planes

Санига

2016

Entropy

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Given a seven-element set X = {1, 2, 3, 4, 5, 6, 7}, there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane-that is to say an unordered triple from X-ordinary or defective, according to whether the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of the order s, 0 ≤ s ≤ 3, if there are s defective lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown that no labeled Fano plane can have all points of zero-th order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out.

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“…Now define K 9,9,9 hexagon triple systems H 1 , H 2 , H 3 , H 4 , H 5 , H 6 as follows: For each i ∈ {1, 2, 3} let (X, • i ) be a commutative quasigroup of order 15 with holes h 1 , h 2 , h 3 , h 4 , and h 5 of size 3. (See [6]. )…”

Section: The Outside 3 · (6k + 3) Constructionmentioning

confidence: 99%

“…(1, 3), (4,2), (2,1), (3,3)], [(5, 3), (8, 2), (3,1), (9, 3), (7, 2), (2, 1)], [(4, 3), (4,1), (1,2), (3,3), (3,1), (2,2)], [(2, 3), (4,1), (3,2), (1,3), (3,1), (4,2)], [(6, 2), (4, 1), (7, 3), (5,2), (3,1), (8, 3)], [(9, 2), (3,1), (6,3), (5,2), (4,1), (5,3)], [(3, 3), (5, 1), (7, 2), (8, 3), (4,1), (4,2)], …”

Section: The Inside Outside Intersection Problem For Hexagon Triple Smentioning

confidence: 99%