Matroids with few non-common bases (Q2488944)

In this note the author revisits theorems of two of his previous papers, namely [Discrete Math. 240, 271--276 (2001; Zbl 0989.05021)] and [Discrete Math. 270, 193--205 (2003; Zbl 1033.05024)]. The hypotheses for many theorems that appear in these papers are significantly weakened but essentially the same conclusions are derived and the same proofs work, so the proofs are not rewritten in the paper under review. The spirit of the paper is captured in the conjecture at the end: Suppose that \(M_1\) and \(M_2\) are vertically \((k+1)\)-connected matroids on the same ground set \(E\) for some positive integer \(k\). If the cardinality of the differece set of the collection of bases of \(M_1\) and \(M_2\) is at most \(k\), then there either is a matroid \(N\) on \(E\) obtained from \(M_2\) by relaxing a sequence of tip-hyperplanes and \(\mathcal{B}(M_1)-\mathcal{B}(M_2)= \mathcal{B}(N)-\mathcal{B}(M_2)\); or both \(M_1\) and \(M_2\) have rank at most \(k\) and do not have disjoint cocircuits.