Non-Monochromatic and Conflict-Free Colorings on Tree Spaces and Planar Network Spaces

It is well known that any set of n intervals in ℝ ^1 admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces.