Matroids with many common bases

AbstractLemos (Discrete Math. 240 (2001) 271–276) proved a conjecture of Mills (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 this paper, we prove a similar result, where the hypothesis of the matroids being k-connected is replaced by the weaker hypothesis of being vertically k-connected