OPEN
This is open, and cannot be resolved with a finite computation.
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].
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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 #838, https://www.erdosproblems.com/838, accessed 2026-01-16
