Problem Statement
Let $F_k(n)$ be minimal such that for any $n$ points in $\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ of the points, and $f_k(n)$ similarly but with lines passing through exactly $k$ points.
Estimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\lim F_k(n)/n^2$ and $\lim f_k(n)/n^2$.
Estimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\lim F_k(n)/n^2$ and $\lim f_k(n)/n^2$.
Categories:
Geometry
Progress
Trivially $f_k(n)\leq F_k(n)$ and $f_2(n)=F_2(n)=\binom{n}{2}$. The problem with $k=3$ is the classical 'Orchard problem' of Sylvester. Burr, Grünbaum, and Sloane [BGS74] have proved that\[f_3(n)=\frac{n^2}{6}-O(n)\]and\[F_3(n)=\frac{n^2}{6}-O(n).\]There is a trivial upper bound of $F_k(n) \leq \binom{n}{2}/\binom{k}{2}$, and hence\[\lim F_k(n)/n^2 \leq \frac{1}{k(k-1)}.\]See also [101].Source: erdosproblems.com/669 | Last verified: January 16, 2026