  We study the stability of theories where the gravitational action has
arbitrary algebraic dependence on the three first traces of the Riemann tensor:
the Ricci tensor, the co-Ricci tensor, and the homothetic curvature tensor. We
collectively call them Ricci-type tensors. We allow arbitrary coupling to
matter. We consider the case when the connection is unconstrained, and the
cases when either torsion or non-metricity is assumed to vanish. We find which
combinations of Ricci-type tensors lead to new degrees of freedom around
Minkowski and FLRW space, and when there are ghosts. None of the theories with
new degrees of freedom are healthy, except the previously known case when
torsion is zero and the action depends only on the Ricci tensor. We find that
projective invariance is not a sufficient condition for a theory to be
ghost-free.
