On the Erdos-Szekeres n-interior point problem

The n-interior point variant of the Erdős-Szekeres problem is the following: for any 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 point sets having at most logarithmic number of convex layers, g(n) exists for all n ≥ 5. We also consider a relaxation of the notion of convex polygons and show that for all n, n ≥ 1, any point set with at least n interior points has an almost convex polygon (simple polygon with at most one concave vertex) that contains exactly n-interior points.