Universal upper bounds on the Bose-Einstein condensate and the Hubbard star

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 |varphirangle with equal amplitudes with respect to the hard-core basis (sites) and when the corresponding N-particle state |Ψrangle 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 |Ψrangle is related to the ground state of a bosonic `Hubbard star' showing Bose-Einstein condensation.