On the independence numbers of a matroid (Q1093645)

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 k \(I^ 2_ k\geq (k+1)I_{k-1}I_{k+1}+I_{k-1}I_ k\). 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 \(k=3\).