 
Summary: Coloring Geometric Range Spaces
Greg Aloupis
Jean Cardinal
S´ebastien Collette§
Stefan Langerman¶
Shakhar Smorodinsky
September 1, 2008
Abstract
We study several coloring problems for geometric rangespaces. In addition to their theoret
ical interest, some of these problems arise in sensor networks. Given a set of points in R2
or
R3
, we want to color them such that every region of a certain family (e.g., every disk containing
at least a certain number of points) contains points of many (say, k) different colors. In this
paper, we think of the number of colors and the number of points as functions of k. Obviously,
for a fixed k using k colors, it is not always possible to ensure that every region containing k
points has all colors present. Thus, we introduce two types of relaxations: either we allow the
number of colors used to increase to c(k), or we require that the number of points in each region
increases to p(k).
Symmetrically, given a finite set of regions in R2
