Polychromatic coloring for half-planes

We prove that for every integer k, every finite set of points in the plane can be k-colored so that every half-plane that contains at least 2k-1 points, also contains at least one point from every color class. We also show that the bound 2k-1 is best possible. This improves the best previously known lower and upper bounds of 43k and 4k-1 respectively. We also show that every finite set of half-planes can be k-colored so that if a point p belongs to a subset H_p of at least 3k-2 of the half-planes then H_p contains a half-plane from every color class. This improves the best previously known upper bound of 8k-3. Another corollary of our first result is a new proof of the existence of small size ε-nets for points in the plane with respect to half-planes.