  The present paper studies existence and distributional uniqueness of
subclasses of stationary hard-core particle systems arising as thinnings of
stationary particle processes. These subclasses are defined by natural
maximality criteria. We investigate two specific criteria, one related to the
intensity of the hard-core particle process, the other one being a local
optimality criterion on the level of realizations. In fact, the criteria are
equivalent under suitable moment conditions. We show that stationary hard-core
thinnings satisfying such criteria exist and are frequently distributionally
unique. More precisely, distributional uniqueness holds in subcritical and
barely supercritical regimes of continuum percolation. Additionally, based on
the analysis of a specific example, we argue that fluctuations in grain sizes
can play an important role for establishing distributional uniqueness at high
intensities. Finally, we provide a family of algorithmically constructible
approximations whose volume fractions are arbitrarily close to the maximum.
