  Suppose that red and blue points occur as independent Poisson processes of
equal intensity in R^d, and that the red points are matched to the blue points
via straight edges in a translation-invariant way. We address several closely
related properties of such matchings. We prove that there exist matchings that
locally minimize total edge length in d=1 and d>=3, but not in the strip R x
[0,1]. We prove that there exist matchings in which every bounded set
intersects only finitely many edges in d>=2, but not in d=1 or in the strip. It
is unknown whether there exists a matching with no crossings in d=2, but we
prove positive answers to various relaxations of this question. Several open
problems are presented.
