On a more general characterisation of Steiner systems
An \((l, k)\)-sparse geometry of cardinality \(n\) is a simple incidence geometry \(\Gamma = ({\mathcal P},{\mathcal B})\) satisfying \(l\leq | B| \leq k < n\), for all \(B\in \mathcal B\). A partial Steiner system \(PS(t, k; n)\), with \(n > k > 1\) and \(k\geq t\geq 1\), is an incidence geometry \(\Gamma = ({\mathcal P},{\mathcal B})\), such that: (i) \(|{\mathcal P}| = n\), (ii) \(| {\mathcal B}| = k\), \(\forall B\in \mathcal B\), (iii) any \(t\)-subset of \(\mathcal P\) is contained in at most one block. If in (iii) any \(t\)-subset of \(\mathcal P\) is contained in exactly one block, then \(\Gamma\) is known as a \textit{Steiner system}, which is denoted by \(S(t, k; n)\). In the paper under review the author introduces the \([k, d]\)-sparse geometries of cardinality \(n\), which are natural generalizations of partial Steiner triple systems \(PS(t, k; n)\), with \(d=2(k-t+1)\), and he gives some characterizations for Steiner systems in terms of sparse geometries.
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