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Embeddings of Steiner triple systems

✍ Scribed by Jean Doyen; Richard M. Wilson

Publisher
Elsevier Science
Year
1973
Tongue
English
Weight
867 KB
Volume
5
Category
Article
ISSN
0012-365X

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✦ Synopsis

If X is a set whose elements are called points and A is a collectioxr of subsets of X (called lines) such that:

(i) any two distinct points of X are contained in exactly one line, (ii) every line contains at least two points, we say that the pair (X, A) is a linear space.

A Steiner triple system is defilaed as a finite nocempty linear space (X, A) all of whose lines are of size 3, i.e., contain exactly 3 points. A Steiner triple system with 1x1 = u is said to be of order u and is denoted by S(u). Kirkman [ 41 has proved that th.ere exist. 4 an S(u) if and only if u 2 1 or 3 (mod 6); any positive integer satisfying this congruence will be called admissible.

If (X, A) and (Y, 23) are two Steiner triple systc':ms such that Y E X and B c A, we :;hall say that (Y, B) is embedded in (or is a subsystem of) (X, A) and that (X, A) contains (Y, 8). If (X, A) is) of order u and (Y, 8) is of order u < u, then u > 2u + 1. Indeed, let p E X -Y. Any line cow taining p has aa: most one point in Y. Therefore tlhere are exactly er lines

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## Abstract 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__ ≑ 1,3 (mod 6), __u__ < __v__ < 2__u__ +  1, we ask for the minimum __r__ such that there exists a

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## Abstract Lindner's conjecture that any partial Steiner triple system of order __u__ can be embedded in a Steiner triple system of order __v__ if $v\equiv 1,3 \; ({\rm mod}\; 6)$ and $v\geq 2u+1$ is proved. Β© 2008 Wiley Periodicals, Inc. J Combin Designs 17: 63–89, 2009