Open Caps and Cups in Planar Point Sets

Points p₁,p₂,...,p_k in the plane, ordered in the x-direction, form a k-cap (k-cup}, respectively) if they are in convex position and p₂,...,p_k-1 lie all above (below, respectively) the segment p₁p_k. We prove the following generalization of the Erdos-Szekeres theorem. For any k, any sufficiently large set P of points in general position contains k points, p₁1,p₂,...,p_k, that form either a k-cap or a k-cup, and there is no point of P vertically above the polygonal line p₁p₂···p_k. We give double-exponential lower and upper bounds on the minimal size of P. We also give several related results.