On Triple Lines and Cubic Curves: The Orchard Problem Revisited

Planar point sets with many triple lines (which contain at least three distinct points of the set) have been studied for 180 years, started with Jackson and followed by Sylvester. Green and Tao (Discret Comput Geom 50(2):409-468, 2013) have shown that the maximum possible number of triple lines for an n-element set is Ln(n-3)/6<SIC> RIGHT FLOOR+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lfloor n\hspace{0.33325pt}(n - 3)/6\rfloor + 1$$\end{document}. Here we address the related problem of describing the structure of the asymptotically near-optimal configurations, i.e., of those for which the number of straight lines which go through three or more points has a quadratic (i.e., best possible) order of magnitude. We pose the problem whether such point sets must always be related to cubic curves. To support this conjecture we settle various special cases; some of them are also related to the four-in-a-line problem of Erdos.