On the Erdős-Szekeres n-interior-point problem

The n-interior-point variant of the Erdős-Szekeres problem is the following: for every n,n≥1, does there exist a g(n) such that every point set in the plane with at least g(n) interior points has a convex polygon containing exactly n interior points. The existence of g(n) has been proved only for n≤3. In this paper, we show that for any fixed r≥2, and for every n≥5, every point set having sufficiently large number of interior points and at most r convex layers contains a subset with exactly n interior points. We also consider a relaxation of the notion of convex polygons and show that for every n,n≥1, any point set with at least n interior points has an almost convex polygon (a simple polygon with at most one concave vertex) that contains exactly n interior points.