Place $n$ points in a general position on the plane. Call a set $S$ of any points stabbing if every triangle formed by the $n$ chosen points contains at least one point from $S$ in its interior. For each $n$ find the smallest number $N$ such that there always exists a stabbing set of $N$ points.
Clearly if the convex hull is a $b$-gon, any triangulation consists of $2n - b - 2$ triangles and we need at least this many points in any stabbing set. When $b = n$ (the chosen points are vertices of a convex polygon), $2n - b - 2 = n - 2$ points are indeed enough. Overall it looks to me that $2n - b - 2$ points might be sufficient.