Two combinatorial problems in the plane (Q1892420)

This paper contains the authors' solution to one problem about arrangements of lines and points in the plane, and a partial solution, due to Dean Hickerson, of another. (The connecting theme is that both problems were posed in a 1978 paper by the same authors.) Let \(t_n\), \(n = 2,3,\dots\), be the number of lines of the arrangement containing exactly \(n\) points; and let \(\varepsilon\) be the lesser of \(\{t_3/ t_2, 1\}\). It is shown that absolute positive constants \(C_1\), \(C_2\) exist such that if the number of points is \(n\), the total number of lines determined by the points is at least \(C_1 \in n^2\); and \(t_3\) is at least \(C_2 \varepsilon^2 n^2\). The second problem asks how small a set \(T\) can be, if there is an \(n\)-point noncollinear set \(S\), disjoint from \(T\), such that every line through two or more points of \(S\) contains a point of \(T\). For \(n \geq 6\), a construction, due to Hickerson, is given for a pair \((S,T)\) such that \(|S| = n\), \(|T| = n-2\).