  We use the recently developed framework of the Mellin bootstrap to study
perturbatively free scalar CFTs in arbitrary dimensions. This approach uses the
crossing-symmetric Mellin space formulation of correlation functions to
generate algebraic bootstrap equations by demanding that only physical
operators contribute to the OPE. We find that there are no perturbatively
interacting CFTs with only fundamental scalars in $d>6$ dimensions (to at least
second order in the perturbation). Our results can be seen as a modest step
towards understanding the space of interacting CFTs in $d>6$ and are consistent
with the intuition that no such CFTs exist.
