Let E(k,β) denote the smallest integer such that any set of at least
E(k,β) points in the plane, no three on a line, contains either an empty
convex polygon with k vertices or an empty pseudo-triangle with β
vertices. The existence of E(k,β) for positive integers k,ββ₯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,β), and prove bounds on E(k,6) and E(6,β), for k,ββ₯3. By
dropping the emptiness condition, we define another related quantity F(k,β), which is the smallest integer such that any set of at least F(k,β) points in the plane, no three on a line, contains a convex polygon with
k vertices or a pseudo-triangle with β vertices. Extending a result of
Bisztriczky and T\'oth [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).Comment: A minor error in the proof of Theorem 2 fixed. Typos corrected. 19
pages, 11 figure