Matroids with few non-common bases

In [On Millsʹs conjecture on matroids with many common bases, Discrete Math. 240 (2001) 271–276], Lemos proved a conjecture of Mills [On matroids with many common bases, Discrete Math. 203 (1999) 195–205]: for two (k+1)(k+1)-connected matroids whose symmetric difference between their collections of bases has size at most kk, there is a matroid that is obtained from one of these matroids by relaxing n1n1 circuit-hyperplanes and from the other by relaxing n2n2 circuit-hyperplanes, where n1n1 and n2n2 are non-negative integers such that n1+n2≤kn1+n2≤k. In [Matroids with many common bases, Discrete Math. 270 (2003) 193–205], Lemos proved a similar result, where the hypothesis of the matroids being kk-connected is replaced by the weaker hypothesis of being vertically kk-connected. In this paper, we extend these results.