PROVED
This has been solved in the affirmative.
- $250
For a set of $n$ points $P\subset \mathbb{R}^2$ let $\ell_1,\ldots,\ell_m$ be the lines determined by $P$, and let $A=\{\lvert \ell_1\cap P\rvert,\ldots,\lvert \ell_m\cap P\rvert\}$.
Let $F(n)$ count the number of possible sets $A$ that can be constructed this way. Is it true that\[F(n) \leq \exp(O(\sqrt{n}))?\]
Erdős writes it is 'easy to see' that this bound would be best possible. This was proved by Szemerédi and Trotter
[SzTr83].
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Additional thanks to: Noga Alon
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 #607, https://www.erdosproblems.com/607, accessed 2026-01-16
