Embeddings on \(S(2,4,v)\)
A Steiner system \(S(2,k,v)\) is a pair \((X,\mathcal A)\) where \(X\) is a \(n\)-set and \(\mathcal A\) is a collection of some \(k\)-subsets (called blocks) of \(X\) such that any two distinct elements of \(X\) are contained in exactly one block in \(\mathcal A\). It has been proved that an \(S(2,4,v)\) exists iff \(v\equiv 1\) or \(4\mod{12}\). Any positive integer satisfying this congruence is called admissible. If \((X,\mathcal A)\) and \((Y,\mathcal B)\) are two systems such that \(X\subseteq Y\) and \({\mathcal A}\subseteq {\mathcal B}\), then \((X,\mathcal A)\) is said to be embedded in \((Y,\mathcal B)\). The necessary condition for an \(S(2,k,u)\) to be embedded in an \(S(2,k,v)\) is that \(n\geq (k-1)m+1\). Several authors have investigated the sufficiency [see \textit{J. Doyen} and \textit{R. M. Wilson}, Discrete Math. 5, 229--239 (1973; Zbl 0263.05017), \textit{A. E. Brouwer} and \textit{H. Lenz}, Contribution to geometry, Proc. Symp., Siegen 1978, 383--389 (1979; Zbl 0425.51006), Eur. J. Comb. 2, 323--330 (1981; Zbl 0477.05016)]. It is proved in this paper that if \(u\) and \(v\) are admissible integers and \(u>85\), \(u\neq 133\), then an \(S(2,4,u)\) can be embedded in some \(S(2,4,v)\) iff \(v\geq 3u+1\). Recently, the sufficiency for the remaining cases when \(u\leq 85\) or \(u=133\) has also been obtained by \textit{R. Rees} and \textit{D. R. Stinson} in their paper ``On the existence of incomplete designs of block size four having one hole'' in Utilitas Math. 35, 119--152 (1989; Zbl 0678.05009).
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