Matroid intersection conjecture in hindrance form

Establish that for every pair of finitary matroids M and N on a common ground set E, if there is no set H that is independent in N but not spanning in N restricted to the M-span of H, then there exists an M-independent set that spans N.

Background

The paper studies the infinite Matroid intersection conjecture of Nash-Williams through the equivalent notion of hindrances. For a pair of matroids (M,N), a hindrance is a set H that is independent in N but fails to span the set spanned by H in M within N. A pair is unhindered when no such set exists.

The conjecture asserts that the absence of hindrances guarantees an M-independent set spanning N. The paper proves a substantial partial result: whenever a common independent set I satisfies r(M/I)<r(N/I), the pair (M,N) is hindered. Thus, the displayed conjecture remains the broader unresolved objective toward which the main theorem advances.

References

The following is known to be equivalent to the Matroid intersection conjecture and it is usually the preferred angle to approach it: If $(M,N)$ is unhindered, then there exists an $M$-independent set that is spanning in $N$.

— Hindrance from a wasteful common independent set  (2501.04342 - Joó, 8 Jan 2025) in Section 1, Introduction; displayed Conjecture (citing Conjecture 3.3 of Aharoni and Ziv)