Before revision:  We also  show that analogous results of the Hadwiger and Debrunner (p,q)-theorem hold in other settings, such as convex sets in \mathbb{R}^d\times\mathbb{Z}^k or abstract convex geometries.
After revision:  We also investigate other conditions on convex polygons for which our algorithm can find a fixed number of points stabbing them. Finally, we show that analogous results of the Hadwiger and Debrunner (p,q)-theorem hold in other settings, such as convex sets in \mathbb{R}^d\times\mathbb{Z}^k or abstract convex geometries.