Disjoint empty convex polygons in planar point sets

Recently, the study of convex polygons has gained a renewed interest because of their importance in computer graphics, geometric learning theory, and artificial intelligence, for instance. Surprisingly, many simple questions are unanswered in this field. Let us start with a beautiful example. We say that a set of points in the plane is in general position if no three of the points lie on a line. Decades ago, Erdős, Klein, and Szekeres posed the problem of determining the maximum number f(k) of points in general position in the plane so that no k points form the vertex set of a convex polygon. Erdős and Szekeres [3] proved that