  There are effective field theories that cannot be embedded in any UV complete
theory. We consider scalar effective field theories, with and without dynamical
gravity, in $D$-dimensional anti-de Sitter (AdS) spacetime with large radius
and derive precise bounds (analytically) on the coupling constants of higher
derivative interactions $\phi^2\Box^k\phi^2$ by only requiring that the dual
CFT obeys the standard conformal bootstrap axioms. In particular, we show that
all such coupling constants, for even $k\ge 2$, must satisfy positivity,
monotonicity, and log-convexity conditions in the absence of dynamical gravity.
Inclusion of gravity only affects constraints involving the
$\phi^2\Box^2\phi^2$ interaction which now can have a negative coupling
constant. Our CFT setup is a Lorentzian four-point correlator in the Regge
limit. We also utilize this setup to derive constraints on effective field
theories of multiple scalars. We argue that similar analysis should impose
nontrivial constraints on the graviton four-point scattering amplitude in AdS.
