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 34k 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 Hp of at least 3k−2 of the half-planes then Hp
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 \eps-nets for points in the plane with
respect to half-planes.Comment: 11 pages, 5 figure