  The cross-couplings among several massless spin-two fields (described in the
free limit by a sum of Pauli-Fierz actions) in the presence of a Dirac field
are investigated in the framework of the deformation theory based on local BRST
cohomology. Under the hypotheses of locality, smoothness of the interactions in
the coupling constant, Poincare invariance, (background) Lorentz invariance and
the preservation of the number of derivatives on each field, we prove that
there are no consistent cross-interactions among different gravitons in the
presence of a Dirac field. The basic features of the couplings between a single
Pauli-Fierz field and a Dirac field are also emphasized.
