Faithful Enclosing of Triple Systems: Doubling the Index

A triple system of order v ≥ 3 and index λ is faithfully enclosed in a triple system of order w ≥ v and index μ ≥ λ when the triples induced on some v elements of the triple system of order w are precisely those from the triple system of order v. When λ = μ, faithful enclosing is embedding; when λ = 0, faithful enclosing asks for an independent set of size v in a triple system of order w. When μ = 2λ, we prove that a faithful enclosing of a triple system of order v and index λ into a triple system of order w and index μ exists if and only if w ≥ d 3v−1 2 e, μ ≡ 0 (mod gcd(w − 2, 6)), and (v, w) 6∈ {(3, 5), (5, 7)}. 1. Background and necessary conditions A triple system of order v and index λ, denoted TS(v, λ), is a pair (V,B). V is a set of v elements, and B is a collection of 3-element subsets of V called triples or blocks. Every 2-subset of V appears in precisely λ of the triples of B. A triple system is simple if it has no repeated blocks. Let T1 = (V,B) be a TS(v, λ) and T2 = (W,D) be a TS(w, μ). T1 is enclosed in T2 if B ⊆ D (where ⊆ is multiset inclusion); T2 is an enclosing of T1. Such an enclosing is faithful when the collection of all triples in D having all three elements from V ⊆W is precisely B. Colbourn, Hamm and Rosa [3] introduced the notions of enclosing and faithful enclosing as a generalization of embedding, which has been widely studied. Faithful enclosing also generalizes the notion of an independent set: taking λ = 0, a TS(v, λ) has no triples, and of course one exists for every v ≥ 0. A faithful enclosing of such a TS(v, 0) in a TS(w, μ) is precisely an independent set of size v in the triple system of order w. Faithful enclosing of triple systems is thus a nontrivial generalization of two apparently different problems: embedding and independent set. In addition, the existence of certain faithful enclosings has applications in the “support size” problem [5]. At the outset, we recall from [2] the necessary conditions for a TS(v, λ) to be faithfully enclosed in a TS(w, μ). We assume that w ≥ v ≥ 3 to avoid trivial cases; we further require that μ ≥ λ ≥ 0, with μ > 0. A necessary condition for a TS(w, μ) to exist is that w ≥ 0, w 6= 2 and μ ≡ 0 (mod gcd(w − 2, 6)); we call such an integer w μ-admissible. This condition is sufficient for the existence of Received February 11, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 05B05, 05B07. 134 D. C. BIGELOW AND C. J. COLBOURN a TS(w, μ); in fact, if in addition μ ≤ w − 2, a simple TS(w, μ) exists [8]. For a faithful enclosing of a TS(v, λ) in a TS(w, μ) to exist, w must be μ-admissible. The faithful nature of the enclosing underpins two further necessary conditions: Lemma 1.1. [2] If a TS(v, λ) is faithfully enclosed in a TS(w, μ), w − v ≥ (μ− λ)(v − 1)/μ, with equality only if w − v is μ-admissible. Lemma 1.2. [2] If a TS(v, λ) is faithfully enclosed in a TS(w, μ), w = v + s, then if 4(μ− λ)v(v − 1) < μ(v + 1), either