Embedding Steiner triple systems into Steiner systems \(S(2,4,v)\). (Q1421522)

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.