  We make an ansatz for the Mellin representation of the four-point amplitude
of half-BPS operators of arbitrary charges at order $\lambda^{-\frac{5}{2}}$ in
an expansion around the supergravity limit. Crossing symmetry and a set of
constraints on the form of the spectrum uniquely fix the amplitude and
double-trace anomalous dimensions at this order. The results exhibit a number
of natural patterns which suggest that the bootstrap approach outlined here
will extend to higher orders in a simple way.
