Maximizing the number of Pasch configurations in a Steiner triple ...

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Maximizing the number of Pasch configurations in a Steiner triple system M. J. Grannell and G. J. Lovegrove Department of Mathematics and Statistics The Open University Walton Hall Milton Keynes MK7 6AA UNITED KINGDOM m.j.grannell@open.ac.uk graham.lovegrove@virgin.net Abstract Let P(v) denote the maximum number of Pasch configurations in any Steiner triple system on v points. It is known that P(v) ≤ M(v) = v(v − 1)(v − 3)/24, with equality if and only if v is of the form 2 n −1. It is also known that limsup v→∞ v=2 n −1 P(v) = 1. We M(v) give a new proof of this result and improved lower bounds on P(v) for certain values of v. AMS classifications: Primary: 05B07. Keywords: Pasch configuration; Steiner triple system. 1 Introduction A Steiner triple system of order v, STS(v), is an ordered pair (V,B) where V isav-elementset(thepoints)andB isasetoftriplesfromV (the blocks), such that each pair from V appears in precisely one block. The necessary and sufficient condition for the existence of an STS(v) is that v ≡ 1 or 3 (mod 6) [6], and these values of v are said to be admissible. We often omit set brackets and commas from triples so that {x,y,z} may be written as xyz when no confusion is likely, and pairs may be treated similarly. 2
A Pasch configuration or quadrilateral is a set of four triples on six distinct points having the form {abc,ade,bdf,cef}. For each admissible v = 7,13, it is known that there is an STS(v) containing no Pasch configurations [4]. At the other extreme, no STS(v) can contain more than M(v) = v(v − 1)(v − 3)/24 Pasch configurations, and this upper bound is only achieved when v = 2 n − 1 and the corresponding STS(v) is the projective system of order 2 n − 1 [7]. Apart from the values v = 2 n − 1, the maximum number of Pasch configurations P(v) in systems of order v is generally unknown. In fact the only known values when v = 2 n −1 appear to be P(9) = 0,P(13) = 13 and P(19) = 84 (for the latter see [2]). How- ever, it was shown by Gray and Ramsay [5] that limsup v→∞ v=2 n −1 P(v) M(v) = 1, and the proof of this rests on some recursive constructions. Various papers, in particular [5, 7], give some lower bounds for P(v) for specific values of v, to which the recursive constructions may be applied to extend these bounds to infinite classes of admissible values v. In this current note we improve the lower bounds for P(v) for certain infinite classes of v. Our bounds provide an alternative direct proof of the asymptotic result mentioned above. The constructions we use might best be described as “Add c” constructions since they can, in suitable circumstances, produce an STS(v +c) from an STS(v) for small values of c. These constructions are not new; the case c = 2 is described in [3]. We start by describing the cases c = 4 and c = 6. A parallel class in an STS(6s+3) is a set of 2s+1 mutually disjoint blocks. 2 Results Suppose we have an STS(6s+3)= (V,B) with two parallel classes P1 and P2 intersecting in a single common block ∞1∞2∞3. Form a regular graph of degree 7, G7, on the vertex set V ′ = V \{∞1,∞2,∞3}. Let ab be an edge of G7 if abc ∈ Pi (i = 1 or 2) or if ab∞i ∈ B (where i = 1,2 or 3). By Vizing’s Theorem, the chromatic index χ ′ of G7 must be either seven or eight. Suppose that χ ′ (G7) = 7. Then each colour class is a one-factor, and so we have seven one factors Fj, j = 1,2,...,7. Now delete those blocks of B that lie in P1 or P2, or contain any of the points ∞1,∞2,∞3, to form the set of blocks B ′ . Then introduce seven new points, say αj, j = 1,2,...,7, and form new blocks by appending αj to each pair ab ∈ Fj. Finally, add seven new blocks forming an STS(7) on the points αj. The new blocks together with the blocks of B ′ form an STS(6s+7) on the point set V ′ ∪{αj : j = 1,2,...,7}. In a similar way an STS(6s +9) might be constructed from an STS(6s+3) having three parallel classes intersecting in a common block. The principal difficulty in applying these constructions 3
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