  We explore the relation between positivity of the energy constraints in
conformal field theories and causality in their dual gravity description. Our
discussion involves CFTs with different central charges whose description, in
the gravity side, requires the inclusion of quadratic curvature corrections. It
is enough, indeed, to consider the Gauss-Bonnet term. We find that both sides
of the AdS/CFT correspondence impose a restriction on the Gauss-Bonnet
coupling. In the case of 6d supersymmetric CFTs, we show the full matching of
these restrictions. We perform this computation in two ways. First by
considering a thermal setup in a black hole background. Second by scrutinizing
the scattering of gravitons with a shock wave in AdS. The different helicities
provide the corresponding lower and upper bounds. We generalize these results
to arbitrary higher dimensions and comment on some hints and puzzles they
prompt regarding the possible existence of higher dimensional CFTs and the
extent to which the AdS/CFT correspondence would be valid for them.
