Problem 669

Erdős Problem #669: declared status 'open'. Formalized: no. 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 $\lim f_k(n)/n^2$. Current best: Burr, Gr\"{u}nbaum, 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]. Prize: no. OEIS: A003035, A006065, A008997. Tags: geometry.