  We perform a thorough study of the theoretical consistency of recently
proposed, viable, quadratic modifications of gravity that are functions of the
the Gauss-Bonnet invariant, regarding the stability of their perturbations
around vacuum, maximally symmetric spaces of constant curvature. We pay special
attention, in particular, to the investigation of pathological instabilities
associated with the occurrence of propagating spin-0 tachyon modes, and with
the development of a graviton ghost. The latter effect is associated with the
known "Ricci stability" issue, well studied in $f(R)$-theories. Within
quadratic modifications of gravity it is discussed for the first time. Special
attention is paid to the requirement of non-negativity of the effective
gravitational coupling, which warrants that the graviton is not a negative-norm
state. It is demonstrated that, several theories that pass the cosmological as
well as the solar system tests, have to be rule out on the basis of the
unavoidable character of these pathological instabilities.
