  Non-linear partially massless (PM) gravity, if it exists, is a theory of
massive gravity in which the graviton has four propagating degrees of freedom.
In PM gravity, a scalar gauge symmetry removes one of the five modes of the
massive graviton. This symmetry ties the value of the cosmological constant to
the mass of the graviton, which in turn can be kept small in a technically
natural way. Thus PM gravity could offer a compelling solution to the old
cosmological constant problem. In this work we look for such a theory among the
known ghost-free massive gravity models with a de Sitter reference metric. We
find that despite the existence of strong supporting evidence for the existence
of a PM theory of gravity, technical obstructions arise which preclude its
formulation using the standard massive gravity framework.
