IMC 2014 · Problem 4

We say that a subset of $\mathbb{R}^n$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-generic if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $k$ and $n$, find the minimal number $d(k,n)$ such that every finite $k$-generic set in $\mathbb{R}^n$ contains a $k$-generic subset with at most $d(k,n)$ elements.