An order-refined and generalized version of the Erdos-Szekeres theorem on convex polygons

The Erdos-Szekeres theorem states that for any natural k there is a natural number g(k) such that any set of at least g(k) points on a plane in general position contains a set of k points that are the extreme points of a convex polytope. We generalize and refine this theorem, having the general-position condition removed and a convex polygon defined as an ordered sequence of points such that the union of the edges of the polygon coincides with the boundary of its convex hull.