For a set of $n$ points $P\subset {\mathbb{R}}^2$ let ${\ell }_1,\dots ,{\ell }_m$ be the lines determined by $P$, and let $A=\{|{\ell }_1\cap P|,\dots ,|{\ell }_m\cap P|\}$.Let $F(n)$ count the number of possible sets $A$ that can be constructed this way. Is it true that$$F(n)\le \exp (O(\sqrt{n}))?$$