  Recent experiments demonstrate the ability to construct cold atom mixtures
with species selective optical lattices. This allows for the possibility of a
mixed-dimension system, where one fermionic atomic species is confined to a two
dimensional lattice, while another species is confined to a three dimensional
lattice that contains the two-dimensional one. We show that by tuning the
density of an arbitrary number of three-dimensional atomic species, we can
engineer an arbitrary, rotationally-symmetric, density-density, effective
interaction for the two-dimensional particles. This possibility allows for an
effective interaction that favours triplet pairing for two-dimensional, $SU(2)$
symmetric particles. Using a functional renormalization-group analysis for the
two-dimensional particles, we derive and numerically confirm that the critical
temperature for triplet pairing depends exponentially on the effective
interaction strength. We then analyse how the stability of this phase is
affected by the particle densities and the fine tuning of interaction
parameters. We conclude by briefly discussing experimental considerations and
the potential to study triplet pairing physics, including Majorana fermions and
spin textures, with cold atoms on optical lattices.
