On sets of sets mutually intersecting in exactly one element
The author studies families \({\mathcal S}\) of subsets of some finite set \(M\) such that each \(S \in {\mathcal S}\) has bounded cardinality \(r\), any two distinct sets of \({\mathcal S}\) intersect in exactly one element and each \(m \in M\) is contained in at least two such subsets. The main result of the paper under review gives an upper bound \(b(r)\) for the cardinality of \({\mathcal S}\) as well as a characterization of such pairs \((M,{\mathcal S})\) that attain this upper bound. In the language of geometry this gives a characterization of finite linear spaces (almost) having the maximum number \(b(r)\) of lines. The author proves that \(b(r) = 2/27r^3 + 7/18 r^2 + 1/2 r + 1/27\). The proof of this result is based on double-counting and a rather complicated and detailed investigation of five different cases. If \({\mathcal L} = (M,{\mathcal S})\) is a linear space with \(|{\mathcal S }|= b(r)\), \(r \neq 4\), then \({\mathcal L}\) is a symmetrical tactical configuration with parameters \((r^2-r/2-1,r)\). If \({\mathcal L} = (M,{\mathcal S})\) is a linear space with \(|{\mathcal S }|= b(r) - 1/27\), \(r \neq 3\), then \({\mathcal L}\) is a symmetrical tactical configuration with parameters \((r^2 - (r+1)/2,r)\). For \(r=7\) an example of such a tactical configuration is known to exist, whereas very little is known about the \((r^2-r/2-1,r)\)-tactical configurations [see \textit{H. Gropp}, Ann. Discrete Math. 52, 227-239 (1992; Zbl 0767.05034)].
scientific article; zbMATH DE number 1443426