  We study the complementary set of a Poissonian ensemble of infinite cylinders
in R^3, for which an intensity parameter u > 0 controls the amount of cylinders
to be removed from the ambient space. We establish a non-trivial phase
transition, for the existence of an unbounded connected component of this set,
as u crosses a critical non-degenerate intensity u*. We moreover show that this
complementary set percolates in a sufficiently thick slab, in spite of the fact
that it does not percolate in any given plane of R^3, regardless of the choice
of u.
