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Embedding Steiner Triple Systems Into Steiner Systems S(2;4;V)

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Embedding Steiner Triple Systems Into Steiner Systems S(2;4;V) Discrete Mathematics 274 (2004) 199–212 www.elsevier.com/locate/disc Embedding Steiner triple systems into Steiner systems S(2; 4;v) Mariusz Meszkaa , Alexander Rosab aDepartment of Applied Mathematics, AGH Krakow, Krakow, Poland bDepartment of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 Received 26 March 2002; received in revised form 13 November 2002; accepted 3 December 2002 Abstract We initiate a systematic study of embeddings of Steiner triple systems into Steiner systems S(2; 4;v). We settle the existence of an embedding of the unique STS(7) and, with one possible exception, of the unique STS(9) into S(2; 4;v). We also obtain bounds for embedding sizes of Steiner triple systems of larger orders. c 2003 Elsevier B.V. All rights reserved. Keywords: Steiner triple systems; Steiner systems S(2; 4;v); Embedding 1. Introduction A Steiner system S(t; k; v) is a pair (V; B) where V is a v-set, v¿0, and B is a collection of k-subsets of V called blocks such that each t-subset of V is contained in exactly one block. A Steiner system S(2; 3;v) is called a Steiner triple system of order v, brie5y STS(v). If we replace in the above deÿnition the words “exactly one” with “at most one”, we obtain the deÿnition of a partial Steiner system S(t; k; v). It is well known that an STS(v) exists if and only if v ≡ 1; 3 (mod 6) [3], and that an S(2; 4;w) exists if and only if w ≡ 1; 4 (mod 12). We refer to these orders as admissible. A parallel class in an STS(v)(V; B) is a set of blocks which partition the set V .AnSTS(v)(V; B)isresolvable if its set of blocks B can be partitioned into par- allel classes. Such a partition is termed resolution.AKirkman triple system KTS(v)of E-mail address: [email protected] (A. Rosa). 0012-365X/03/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0012-365X(03)00089-X 200 M. Meszka, A. Rosa / Discrete Mathematics 274 (2004) 199–212 order v is an STS(v) together with a particular resolution R. It is well known that a resolvable STS(v) [and thus a KTS(v)] exists if and only if v ≡ 3 (mod 6) [3]. A Steiner system S(t; k; v)(V; B)isembedded in a Steiner system S(t;k;w)(W; C) if V ⊂ W , and C|V = B, i.e. B = {B : B ∈ C;B ⊂ V }. In this case we also say that (W; C) contains (V; B) as a subsystem. The best studied examples of embeddings of Steiner systems are those when t=t =2 or 3 and k = k = 3 or 4. The following two well-known theorems provide a deÿnite answer in two of the cases. The Doyen–Wilson Theorem (Doyen and Wilson [5]). An STS(v) can be embedded in an STS(w) if and only if v = w or w ¿ 2v +1, and both v; w are admissible. The Rees–Stinson Theorem (The Rees and Stinson [16]). A Steiner system S(2; 4;v) can be embedded in a Steiner system S(2; 4;w) if and only if v = w or w ¿ 3v +1 and both v; w are admissible. Hartman [7] has made substantial progress towards proving the conjecture that a Steiner system S(3; 4;v) (also called Steiner quadruple system) can be embedded into a Steiner system S(3; 4;w) if and only if w ¿ 2v and v; w ≡ 2; 4 (mod 6) [or v = w,of course]. However, this conjecture remains open. Much less appears known in the case when t =t but k¡k. Of course, there are the well-known geometric examples obtained from embedding aHne planes in projective planes, or, in our notation, embedding Steiner systems S(2;q;q2) into Steiner systems S(2;q+1;q2 + q + 1), and, more generally, embedding aHne spaces into projective spaces. Other examples include examination of projective embeddings of small Steiner triple systems by Limbos [11], and embeddings of aHne and projective spaces in projective planes [8]. In this paper we concentrate on the case when t = t =2;k =3;k = 4, i.e. on the question of embeddings of Steiner triple systems into Steiner systems S(2; 4;v). To the best of our knowledge, this case has not been studied systematically, although sporadic results concerning this case can be found in the sources cited already, and also in [17], and [9]; in the latter article, one ÿnds examples of STS(7) embedded in S(2; 4; 25). To see that the general question “Which Steiner triple systems can be embedded in a Steiner system S(2; 4;v)?” is not easy and is not likely to be settled in one fell swoop, it suHces to realize that this question includes, as a special case, the question of de Resmini [17] about the existence of the “century design”, as well as some additional unsolved problems related to specialized colourings of Steiner systems S(2; 4;v) with blocks having prescribed colour patterns [13]. There is another feature which distinguishes this problem from embedding problems when t = t;k = k: there is no replacement property. If we have an embedding of, say, an STS(v) into an STS(w), then any other STS(v) can also be embedded in an STS(w). This is no longer true when one considers embeddings where k¡k, e.g. those of STSs into S(2; 4;v). In this paper, we settle the existence of an embedding of the unique STS(7) (i.e. the projective plane of order 2, or Fano plane) and, with one possible exception, of M. Meszka, A. Rosa / Discrete Mathematics 274 (2004) 199–212 201 the unique STS(9) (i.e. the aHne plane of order 3) into Steiner systems S(2; 4;w): an S(2; 4;w) containing STS(7) exists if and only if w ¿ 25;w ≡ 1; 4 (mod 12), while an S(2; 4;w) containing STS(9) exists if and only if w =13 or w ¿ 28, w ≡ 1; 4 (mod 12), except possibly when w = 37. We also obtain bounds for embedding sizes of Steiner triple systems of larger orders. 2. Preliminaries and necessary conditions In addition to the Rees–Stinson Theorem given in the Introduction, we will also make use of the following theorems. Ganter’s Theorem (Ganter [6]). Every partial Steiner system S(2;k;v) can be embed- ded in some Steiner system S(2;k;w). However, we must note that for k¿3 the order w of the containing Steiner system is, in general, exponential in v. Theorem 1 (Mullin et al. [14] and Colbourn and Rosa [3]). A resolvable STS(2v+1) containing an STS(v) exists when v ≡ 1 (mod 6), except possibly if v ∈{115; 145; 205; 265; 355; 415; 649; 697}. Theorem 2 (Rees and Stinson [15]). A KTS(v) can be embedded in a KTS(w) if and only if w = v or w ¿ 3v, w ≡ v ≡ 3 (mod 6). Let (V; B)beanSTS(v). Deÿne E(V; B), the embedding spectrum for (V; B), as follows: E(V; B)={w: there exists an S(2; 4;w) containing (V; B) as a subsystem}. Then E(v), the embedding spectrum for v, is deÿned as E(v)= E(V; B), where the union is taken over all STS(v)(V; B). For an STS(v)(V; B), deÿne further m(V; B) = min E(V; B), m(v) = min E(v), and q(V; B) = min{q: there exists an S(2; 4;w) containing (V; B) as a subsystem for all admissible w ¿ q}. The quantity q(v) is deÿned similarly: q(v) = min{q: there exists an S(2; 4;w) containing some STS(v) as a subsystem for all admissible w ¿ q}. The following is an easy consequence of Ganter’s Theorem. Theorem 3. Every Steiner triple system S=(V; B) can be embedded in some S(2; 4;w). Proof. It suHces to convert the STS S into a partial Steiner system S(2; 4;v). This can be done in many ways, one of which is the following. Let C ={c1;:::;ct} be the set of block colour classes (a block colour class is the set of blocks coloured with the same 202 M. Meszka, A. Rosa / Discrete Mathematics 274 (2004) 199–212 colour) in any block colouring of S. Then form a partial Steiner system S(2; 4;v)on the set V ∪ C by extending each triple T of the colour class ci to a 4-subset T ∪{ci}. The rest follows from Ganter’s Theorem. Thus the sets E(V; B) are nonempty for all STS (V; B), and so are, of course, the sets E(v) for all admissible v. Furthermore, m(V; B), q(V; B), m(v) and q(v) are all well deÿned. Lemma 4. For all v ≡ 1; 3 (mod 6), q(v) 6 3m(v)+1. Proof. This follows directly from the Rees–Stinson Theorem. Lemma 5. For all STS(v)(V; B), m(V; B) ¿ (3v − 1)=2. Proof. Suppose an STS(v)(V; B) is embedded as a subsystem in an S(2; 4;w)(W; C). Every element of V appears r =(v − 1)=2 times in B, and every element of W must appear r =(w − 1)=3 times in C.
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