  It has been pointed out that non-singular cosmological solutions in
second-order scalar-tensor theories generically suffer from gradient
instabilities. We extend this no-go result to second-order gravitational
theories with an arbitrary number of interacting scalar fields. Our proof
follows directly from the action of generalized multi-Galileons, and thus is
different from and complementary to that based on the effective field theory
approach. Several new terms for generalized multi-Galileons on a flat
background were proposed recently. We find a covariant completion of them and
confirm that they do not participate in the no-go argument.
