  Random sets with long-range dependence can be generated using a Boolean model
with power-law grain sizes. We study thinnings of such Boolean models which
have the hard-core property that no grains overlap in the resulting germ-grain
model. A fundamental question is whether long-range dependence is preserved
under such thinnings. To answer this question we study four natural thinnings
of a Poisson germ-grain model where the grains are spheres with a regularly
varying size distribution. We show that a thinning which favors large grains
preserves the slow correlation decay of the original model, whereas a thinning
which favors small grains does not. Our most interesting finding concerns the
case where only disjoint grains are retained, which corresponds to the
well-known Mat\'ern type I thinning. In the resulting germ-grain model, typical
grains have exponentially small sizes, but rather surprisingly, the long-range
dependence property is still present. As a byproduct, we obtain new mechanisms
for generating homogeneous and isotropic random point configurations having a
power-law correlation decay.
