OPEN
Let $f(n)$ be maximal such that any $n$ points in $\mathbb{R}^2$, with no three on a line, determine at least $f(n)$ different convex subsets. Estimate $f(n)$ - in particular, does there exist a constant $c$ such that
\[\lim \frac{\log f(n)}{(\log n)^2}=c?\]
A question of Erdős and Hammer. Erdős proved in
[Er78c] that there exist constants $c_1,c_2>0$ such that
\[n^{c_1\log n}<f(n)< n^{c_2\log n}.\]
See also [107].
