On the Existence of a Point Subset with Three or Five Interior Points

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer $k\ge1$, let $h(k)$ be the smallest integer such that every point set in the plane, no three collinear, with at least $h(k)$ interior points, has a subset with $k$ or $k+2$ interior points of $P$. We prove that $h(3)=8$.