  For $N$ hard-core bosons on an arbitrary lattice with $d$ sites and
independent of additional interaction terms we prove that the hard-core
constraint itself already enforces a universal upper bound on the Bose-Einstein
condensate given by $N_{max}=(N/d)(d-N+1)$. This bound can only be attained for
one-particle states $|\varphi\rangle$ with equal amplitudes with respect to the
hard-core basis (sites) and when the corresponding $N$-particle state
$|\Psi\rangle$ is maximally delocalized. This result is generalized to the
maximum condensate possible within a given sublattice. We observe that such
maximal local condensation is only possible if the mode entanglement between
the sublattice and its complement is minimal. We also show that the maximizing
state $|\Psi\rangle$ is related to the ground state of a bosonic `Hubbard star'
showing Bose-Einstein condensation.
