  We consider the four-point correlator of the stress tensor multiplet in N=4
SYM. We construct all solutions consistent with crossing symmetry in the limit
of large central charge c ~ N^2 and large g^2 N. While we find an infinite
tower of solutions, we argue most of them are suppressed by an extra scale
\Delta_{gap} and are consistent with the upper bounds for the scaling dimension
of unprotected operators observed in the numerical superconformal bootstrap at
large central charge. These solutions organize as a double expansion in 1/c and
1/\Delta_{gap}. Our solutions are valid to leading order in 1/c and to all
orders in 1/\Delta_{gap} and reproduce, in particular, instanton corrections
previously found. Furthermore, we find a connection between such upper bounds
and positivity constraints arising from causality in flat space. Finally, we
show that certain relations derived from causality constraints for scattering
in AdS follow from crossing symmetry.
