SOLVED
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 $n\geq 7$.
This problem is #2 in Ramsey Theory in the graphs problem collection.
