  We investigate, in any spacetime dimension >=3, the problem of consistent
couplings for a finite collection of massless, spin-2 fields described, in the
free limit, by a sum of Pauli-Fierz actions. We show that there is no
consistent (ghost-free) coupling, with at most two derivatives of the fields,
that can mix the various "gravitons". In other words, there are no
Yang-Mills-like spin-2 theories. The only possible deformations are given by a
sum of individual Einstein-Hilbert actions. The impossibility of
cross-couplings subsists in the presence of scalar matter. Our approach is
based on the BRST-based deformation point of view and uses results on the
so-called "characteristic cohomology" for massless spin-2 fields which are
explained in detail.
