On the Erdos-Szekeres n-interior point problem

The n-interior point variant of the Erdos-Szekeres problem is to show the following: For any n, n ⩾ 1 , every point set in the plane with sufficient number of interior points contains a convex polygon containing exactly n-interior points. This has been proved only for n ⩽ 3 . In this paper, we prove it for pointsets having atmost logarithmic number of convex layers. We also show that any pointset containing atleast n interior points, there exists a 2-convex polygon that contains exactly n-interior points.