  Hard core bosons in a large class of one or two dimensional flat band systems
have an upper critical density, below which the ground states can be described
completely. At the critical density, the ground states are Wigner crystals. If
one adds a particle to the system at the critical density, the ground state and
the low lying multi particle states of the system can be described as a Wigner
crystal with an additional pair of particles. The energy band for the pair is
separated from the rest of the multi-particle spectrum. The proofs use a
Gerschgorin type of argument for block diagonally dominant matrices. In certain
one-dimensional or tree-like structures one can show that the pair is
localised, for example in the chequerboard chain. For this one-dimensional
system with periodic boundary condition the energy band for the pair is flat,
the pair is localised.
