  For each positive integer $n \geq 4$, we give an inequality satisfied by rank
functions of arrangements of $n$ subspaces. When $n=4$ we recover Ingleton's
inequality; for higher $n$ the inequalities are all new. These inequalities can
be thought of as a hierarchy of necessary conditions for a (poly)matroid to be
realizable. Some related open questions about the "cone of realizable
polymatroids" are also presented.
