Mathematics – Combinatorics
Scientific paper
2009-05-11
J. Combin. Theory Ser. A, 2010
Mathematics
Combinatorics
10 pages, comments welcome. v2: correction to proof of Prop. 3, improved "Future directions" section, other minor improvements
Scientific paper
10.1016/j.jcta.2009.10.014
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.
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