  Motivated by the recent discoveries of superconductivity in bilayer and
trilayer graphene, we theoretically investigate superconductivity and other
interaction-driven phases in multilayer graphene stacks. To this end, we study
the density of states of multilayer graphene with up to four layers at the
single-particle band structure level in the presence of a transverse electric
field. Among the considered structures, tetralayer graphene with rhombohedral
(ABCA) stacking reaches the highest density of states. We study the phases that
can arise in ABCA graphene by tuning the carrier density and transverse
electric field. For a broad region of the tuning parameters, the presence of
strong Coulomb repulsion leads to a spontaneous spin and valley symmetry
breaking via Stoner transitions. Using a model that incorporates the
spontaneous spin and valley polarization, we explore the Kohn-Luttinger
mechanism for superconductivity driven by repulsive Coulomb interactions. We
find that the strongest superconducting instability is in the $p-$wave channel,
and occurs in proximity to the onset of Stoner transitions. Interestingly, we
find a range of densities and transverse electric fields where
superconductivity develops out of a strongly corrugated, singly-connected Fermi
surface in each valley, leading to a topologically non-trivial chiral $p+ip$
superconducting state with an even number of co-propagating chiral Majorana
edge modes. Our work establishes ABCA stacked tetralayer graphene as a
promising platform for observing strongly correlated physics and topological
superconductivity.
