  We report yes-go and no-go results on consistent cross-couplings for a
collection of gravitons. Motivated by the search of theories where multiplets
of massless spin-two fields cross-interact, we look for all the consistent
deformations of a positive sum of Pauli-Fierz actions. We also investigate the
problem of deforming a (positive and negative) sum of linearized Weyl gravity
actions and show explicitly that there exists multi-Weyl-graviton theories. As
the single-graviton Weyl theory, these theories do not have an energy bounded
from below.
