Deterministic Thinning of Finite Poisson Processes

Let Π and Γ 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 Π and Γ such that Γ is a deterministic function of Π, and all points of Γ are points of Π. 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 Π than in Γ.