DISPROVED
This has been solved in the negative.
Let $g(k)$ be the smallest integer (if any such exists) such that any $g(k)$ points in $\mathbb{R}^2$ contains an empty convex $k$-gon (i.e. with no point in the interior). Does $g(k)$ exist? If so, estimate $g(k)$.
A variant of the 'happy ending' problem
[107], which asks for the same without the 'no point in the interior' restriction. Erdős observed $g(4)=5$ (as with the happy ending problem) but Harborth
[Ha78] showed $g(5)=10$. Nicolás
[Ni07] and Gerken
[Ge08] independently showed that $g(6)$ exists. Horton
[Ho83] showed that $g(n)$ does not exist for $n\geq 7$.
Heule and Scheucher
[HeSc24] have proved that $g(6)=30$.
This problem is
#2 in Ramsey Theory in the graphs problem collection.
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This page was last edited 30 December 2025.
Additional thanks to: Boris Alexeev and Zach Hunter
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 #216, https://www.erdosproblems.com/216, accessed 2026-01-16
