Faithful Enclosing of Triple Systems: A Generalization of Stern’s Theorem

A triple system of order v 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. A generalization of Stern’s theorem for embedding is proved: A triple system of order v and index λ can be faithfully enclosed in a triple system of order w and index μ whenever w ≥ 2v+1, μ ≥ λ and μ ≡ 0 mod gcd(w − 2,6).