Embedding Steiner triple systems into Steiner systems \(S(2,4,v)\).
A Steiner system \(S(t,k,v)\) \((V,{\mathcal B})\) is embedded in a Steiner system \(S(t',k',w)\) \((W,{\mathcal C})\) if \(V \subset W\) and \({\mathcal C}| V = {\mathcal B}\). So far the best studied examples of embeddings of Steiner systems are those when \(t = t' = 2\) or \(3\) and \(k = k' = 3\) or \(4\). In this paper the authors concentrate on the question of embeddings of Steiner triple systems into Steiner systems \(S(2,4,v)\) (i.e. \(t = t' = 2\), \(k = 3\) and \(k' = 4\)). They settle the existence of embeddings of the unique \(\text{STS} (7)\) and, with one possible exception, of the unique \(\text{STS} (9)\). In particular they show that an \(\text{STS} (2,4,w)\) containing an \(\text{STS} (7)\) exists if and only if \(w \geq 25\), \(w \equiv 1,4 \pmod{12}\), and that an \(\text{STS} (2,4,w)\) containing an \(\text{STS} (9)\) exists if and only if \(w = 13\) or \(w \geq 28\), \(w \equiv 1,4 \pmod{12}\), except possibly when \(w = 37\). They also obtain bounds for embedding sizes of Steiner triple systems of larger orders. They conclude by mentioning some open problems, some of which relate to outstanding colouring problems.
scientific article; zbMATH DE number 2032862
scientific article