  In this paper, we analyze the constraints imposed by unitarity and crossing
symmetry on conformal theories in large dimensions. In particular, we show that
in a unitary conformal theory in large dimension $D$, the four-point function
of identical scalar operators $\phi$ with scaling dimension $\Delta_\phi$ such
that $\Delta_\phi/D<3/4$, is necessarily that of the generalized free field
theory. This result follows only from crossing symmetry and unitarity. In
particular, we do not impose the existence of a conserved spin two operator
(stress tensor). We also present an argument to extend the applicability of
this result to a larger range of conformal dimensions, namely to
$\Delta_\phi/D<1$. This extension requires some reasonable assumptions about
the spectrum of light operators. Together, these results suggest that if there
is a non-trivial conformal theory in large dimensions, not necessarily having a
stress tensor, then its relevant operators must be exponentially weakly coupled
with the rest.
