New inequalities for subspace arrangements

For each positive integer n>=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.

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