Matroid analogue of simultaneous flag generation

Determine whether, for every rank-\(d\) matroid \(M\) and every \(m\)-tuple of complete flags of flats in \(M\), there exists a set of \(\mu(m,d)\) hyperplanes whose intersections generate all the flats in all the flags; if not, determine the worst-case number of hyperplanes as a function of \(m\) and \(d\).

Background

The paper develops a dualized matroid formulation of the vector-space problem: given complete flags of flats, one seeks a collection of hyperplanes whose selected intersections produce every flat in each flag. This formulation is equivalent to the vector-space formulation when the matroid arises from a projective vector space.

The authors explain that the two-flag analogue and a version of the key triple-intersection lemma extend to matroids, while another crucial lemma fails for general matroids because the ground set may be the union of two proper flats. The upper-bound argument does extend to unbreakable matroids, but the general matroid version remains unresolved. The question therefore asks both whether the vector-space bound μ(m,d)\mu(m,d) remains valid universally and, if it does not, what the correct worst-case bound should be.

References

Let m,d \in \mathbb{N}$. Is it true that for every rank-$d$ matroid $M$ and every $m$-tuple of complete flags of flats in $M$, there is a set of $\mu(m,d)$ hyperplanes in $M$ that generates (via intersections) all of the flags? If not, then how many hyperplanes (as a function of $m,d$) are needed in the worst case?

— Simultaneous generating sets for flags  (2502.09530 - Glaudo et al., 13 Feb 2025) in Question in Section 5.3, “Matroid analogues”