  It has recently been argued that there may be a nontrivial four-dimensional
limit of the higher-dimensional Gauss--Bonnet and Lovelock interactions and
that this might provide a loophole allowing for new four-dimensional
gravitational theories, possibly without a standard Lagrangian. We investigate
this claim by studying tree-level graviton scattering amplitudes, allowing us
to draw conclusions independently of the Lagrangian. By taking four-dimensional
limits of higher-dimensional scattering amplitudes of the Gauss--Bonnet theory,
we find four-dimensional amplitudes that are different from general relativity;
however, these amplitudes are not new since they all come from certain
scalar-tensor theories. The nontrivial limit that does not lead to infinite
strong coupling around flat space leads to $(\partial\phi)^4$ theory. We argue
that there cannot be any six-derivative purely gravitational four-point
amplitudes in any dimension other than those coming from Lovelock theory by
directly constructing the on-shell amplitudes. In particular, there can be no
new such amplitudes in four dimensions beyond those of general relativity. We
also present some new results on the spin-averaged cross section for
graviton-graviton scattering in general relativity and Gauss--Bonnet theory in
arbitrary dimensions.
