  We initiate a systematic enumeration and classification of entropy
inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory
states with smooth holographic dual geometries. For 2, 3, and 4 regions, we
prove that the strong subadditivity and the monogamy of mutual information give
the complete set of inequalities. This is in contrast to the situation for
generic quantum systems, where a complete set of entropy inequalities is not
known for 4 or more regions. We also find an infinite new family of
inequalities applicable to 5 or more regions. The set of all holographic
entropy inequalities bounds the phase space of Ryu-Takayanagi entropies,
defining the holographic entropy cone. We characterize this entropy cone by
reducing geometries to minimal graph models that encode the possible cutting
and gluing relations of minimal surfaces. We find that, for a fixed number of
regions, there are only finitely many independent entropy inequalities. To
establish new holographic entropy inequalities, we introduce a combinatorial
proof technique that may also be of independent interest in Riemannian geometry
and graph theory.
