Holes or Empty Pseudo-Triangles in Planar Point Sets

Let E(k, l) denote the smallest integer such that any set of at least E(k, l) points in the plane, no three on a line, contains either an empty convex polygon with k vertices or an empty pseudo-triangle with l vertices. The existence of E(k, l) for positive integers k, l ≥ 3, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565–576, 2007]. In this paper, following a series of new results about the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of E(k, 5) and E(5, l), and prove bounds on E(k, 6) and E(6, l), for k, l ≥ 3. By dropping the emptiness condition, we define another related quantity F (k, l), which is the smallest integer such that any set of at least F (k, l) points in the plane, no three on a line, contains a convex polygon with k vertices or a pseudo-triangle with l vertices. Extending a result of Bisztriczky and Tóth [Discrete Geometry, Marcel Dekker, 49–58, 2003], we obtain the exact values of F (k, 5) and F (k, 6), and obtain non-trivial bounds on F (k, 7).