  Let Pi and Gamma be homogeneous Poisson point processes on a fixed set of
finite volume. We prove a necessary and sufficient condition on the two
intensities for the existence of a coupling of Pi and Gamma such that Gamma is
a deterministic function of Pi, and all points of Gamma are points of Pi. The
condition exhibits a surprising lack of monotonicity. However, in the limit of
large intensities, the coupling exists if and only if the expected number of
points is at least one greater in Pi than in Gamma.

Probability