On sets defining few ordinary lines (Q368773)

In this nice paper, the authors solve asymptotically two classical problems concerning finite configuration of points in the plane. The first main result is related to the well-known Sylvester-Gallai theorem. If \(P\) is a given set of \(n\) points in the plane then the ordinary line is defined to be a line containing exactly two points of \(P\). The authors prove that if \(n\) is large, then there are at least \(n/2\) ordinary lines confirming for large \(n\) the Dirac-Motzkin conjecture. (We note here that for some small values of \(n\) there are known configurations with \((n-1)/2\) ordinary lines for e.g. \(n=7\).) The second main result of this paper is related to the ``orchard-planting problem''. This problem concerns the \(3\)-rich lines i.e. the lines that contain precisely \(3\) points of \(P\). For large \(n\), the authors determine the maximum number of the \(3\)-rich lines. Underlying the proof of the above results, the authors prove a structure theorem which states that if \(P\) has at most \(Kn\) ordinary lines then all but \(O(K)\) points of \(P\) lie on a cubic curve, if \(n\) is sufficiently large depending on \(K\).