Linear spaces and partitioning the projective plane
Let \({\mathcal P}=(P,\mathcal L)\) be a finite projective plane of order \(n\). Let \(Q\subset P\) be of cardinality at most \(2n-1\) and let \(\Pi\subset\mathcal L\) with \(|\Pi|<n+1+\sqrt n\). \textit{K. Metsch} [Lect. Notes Math. 1490, Springer, Berlin (1991; Zbl 0744.51005)] proved that the lines of \(\Pi\) are concurrent, if no point row (i.e. all points incident with some line) of \(\Pi\) is contained in \(Q\) and every point outside \(Q\) lies on exactly one line of \(\Pi\). It is easy to see that the bound on \(|Q|\) is almost sharp by considering a plane \(\mathcal P\) which possesses a Baer subplane, i.e. a subplane of order \(\sqrt n\). This gives \(Q\) with \(|Q|=2n+1\). The author proves that in the case \(|Q|=2n\) Metsch's result is still valid with the exception of one single configuration.
scientific article; zbMATH DE number 1076900