On the independence numbers of a matroid

AbstractGiven a finite subset E of a vector space of dimension 4, the number of k-independent subsets of E will be denoted by Ik. We prove that kIk2 ≥ (k + 1)Ik − 1Ik + 1 + Ik − 1Ik. 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