On the Erdős–Szekeres -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.