OPEN
This is open, and cannot be resolved with a finite computation.
Let $r,k\geq 2$ be fixed. Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such that if there are at least $f_{r,k}(n)$ many ordinary lines (lines containing exactly two points) then there is a set $A'\subseteq A$ of $r$ points such that all $\binom{r}{2}$ many lines determined by $A'$ are ordinary.
Is it true that $f_{r,k}(n)=o(n^2)$, or perhaps even $\ll n$?
Turán's theorem implies\[f_{r,k}(n) \leq \left(1-\frac{1}{r-1}\right)\frac{n^2}{2}+1.\]See also
[209].
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #960, https://www.erdosproblems.com/960, accessed 2025-11-16
