Matroids with few non-common bases

In On Mills's 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)-connected matroids whose symmetric difference between their collections of bases has size at most k, there is a matroid that is obtained from one of these matroids by relaxing n₁ circuit-hyperplanes and from the other by relaxing n₂ circuit-hyperplanes, where n₁ and n₂ are non-negative integers such that n₁+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 k-connected is replaced by the weaker hypothesis of being vertically k-connected. In this paper, we extend these results.