Lines, line-point incidences and crossing families in dense sets
A set \(P\) of \(n\) points in the plane is called \(\alpha\)-dense if the ratio between the maximum and the minimum distance in \(P\) is at most \(\alpha\sqrt n\). By a crossing family we mean a set of line segments such that the relative interiors of every two of them intersect. The author proves that if a set \(P\) on \(n\) points is \(\alpha\)-dense, where \(\alpha\geq\sqrt {2\sqrt 3}/ \pi\), then there exists a crossing family with end-points in \(P\) whose size is \(\Omega(n (\log n)^{-4\log \log\log n+c})\), where \(c=4\log (1000\alpha^2+8)\) and where the base of logarithms is 2.
scientific article; zbMATH DE number 936877