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).