Problem 588

Let $f_k(n)$ be minimal such that if $n$ points in $\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many lines containing at least $k$ points. Is it true that

f_k(n)=o(n^2)

for $k\geq 4$ ?

Source: erdosproblems.com/588

Geometry

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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.
Ka63: F. Kárteszi, Sylvester egy tételéről és Erdős egy sejtéséről. Matematikai Lapok (1963), 3-10.
SoSt13: Solymosi, József and Stojaković, Miloš, Many collinear $k$ -tuples with no $k+1$ collinear points. Discrete Comput. Geom. (2013), 811-820.

Problem 588

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