Problem 101

Given $n$ points in $\mathbb{R}^2$ , no five of which are on a line, the number of lines containing four points is $o(n^2)$ .

Status

Open

No claim settles this problem.

BGS74: Burr, Stefan A. and Grünbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.
FuPa84: Füredi, Z. and Palásti, I., Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. (1984), 561-566.
Gr76: Grünbaum, Branko, New views on some old questions of combinatorial geometry. Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo I (1976), 451-468.
SoSt13: Solymosi, József and Stojaković, Miloš, Many collinear $k$ -tuples with no $k+1$ collinear points. Discrete Comput. Geom. (2013), 811-820.