On the independence numbers of a matroid
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Abstract Given a finite subset E of a vector space of dimension 4, the number of k -independent subsets of E will be denoted by I k . We prove that kI k 2 ≥ ( k + 1) I k − 1 I k + 1 + I k − 1 I k . The equality holds if and only if all 4-subsets of E are independent. We prove this relation for matroids of rank 4. In particular we prove Mason's conjecture on the independence numbers of a matroid for k =3.
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