  Given a geometric hypergraph (or a range-space) $H=(V,\cal E)$, a coloring of
its vertices is said to be conflict-free if for every hyperedge $S \in \cal E$
there is at least one vertex in $S$ whose color is distinct from the colors of
all other vertices in $S$. The study of this notion is motivated by frequency
assignment problems in wireless networks. We study the list-coloring (or
choice) version of this notion. In this version, each vertex is associated with
a set of (admissible) colors and it is allowed to be colored only with colors
from its set. List coloring arises naturally in the context of wireless
networks.
  Our main result is a list coloring algorithm based on a new potential method.
The algorithm produces a stronger unique-maximum coloring, in which colors are
positive integers and the maximum color in every hyperedge occurs uniquely. As
a corollary, we provide asymptotically sharp bounds on the size of the lists
required to assure the existence of such unique-maximum colorings for many
geometric hypergraphs (e.g., discs or pseudo-discs in the plane or points with
respect to discs). Moreover, we provide an algorithm, such that, given a family
of lists with the appropriate sizes, computes such a coloring from these lists.
