  We show that there can be no new Lorentz invariant kinetic interactions free
from the Boulware-Deser ghost in four dimensions in the metric formulation of
gravity, beyond the standard Einstein-Hilbert, up to total derivatives. We use
dimensional deconstruction as a way to motivate a non-linear ansatz for
potential new ghost free kinetic interactions for massive gravity, bi-gravity
and multi-gravity in four and higher dimensions. These interactions descend
from Lovelock terms, and so naively one might expect the interactions to be
ghost free. However we show that these new interactions inevitably lead to more
than five propagating degrees of freedom. We then perform a general
perturbative analysis in four dimensions, and show that the only term with two
derivatives that does not introduce a ghost is the Einstein-Hilbert term. This
result extends to all orders in perturbations.
