  What is the space of weakly-coupled, gravitational theories which contain
massive, higher-spin particles? This class of theories is highly constrained
and it is conjectured their ultraviolet completion must be string theory. We
provide more evidence for this conjecture by studying the Regge limit in large
$N$, $4d$ CFTs with single-trace operators of unbounded spin. We show that in
the Regge limit, these theories have bulk scattering amplitudes which are
consistent with the string theory prediction to all orders in $1/N$ for large,
but finite, coupling. In the language of Regge theory, we show Pomeron exchange
naturally exponentiates in the $1/N$ expansion. To do this, we solve the
bootstrap equations at tree-level and then use the Lorentzian inversion formula
to find the one-loop correlator in the Regge limit. This is a unitarity method
for AdS/CFT which can be repeated iteratively to make all orders statements. We
also explain under what conditions the tree-level result exponentiates in the
$1/N$ expansion at arbitrary coupling. Finally, we comment on further inelastic
effects and show they give subleading contributions at large coupling. As a
consistency check, we recover results from bulk Einstein gravity in the limit
where all higher-spin particles decouple.
