IMC 2014 Problem 9

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 $(k, n)$ , find the minimal number of $d(k, n)$ such that every finite $k$ -generic set in $\mathbb{R}^n$ contains a $k$ -generic subset with at most $d(k, n)$ elements.

(Proposed by Shachar Carmeli, Weizmann Inst. and Lev Radzivilovsky, Tel Aviv Univ.)