Let E be a ground set, let mathfrakF(E) denote the relevant class of matroids on E, and for j∈0,1 let I_M_j be the collection of independent sets of M_j. A set I spans a set X in a matroid if X⊆operatornamespan(I). Nash-Williams's matroid intersection conjecture. For every M_0,M_1∈mathfrakF(E), there is an I∈I_M_0capI_M_1 and a partition E=E_0 sqcup E_1 such that I cap E_i spans E_i in M_i for i∈0,1. This is an infinite-matroid analogue of matroid intersection, and the paper discusses it as an application of its main Cantor–Bernstein-type results. Its resolution is not established in the supplied text.

Nash-Williams's matroid intersection conjecture

About 15 years old · traced to 2011

Let $E$ be a ground set, let $\mathfrak{F}(E)$ denote the relevant class of matroids on $E$ , and for $j\in\{0,1\}$ let $\mathcal{I}_{M_j}$ be the collection of independent sets of $M_j$ . A set $I$ spans a set $X$ in a matroid if $X\subseteq\operatorname{span}(I)$ . Nash-Williams's matroid intersection conjecture. For every $M_0,M_1\in\mathfrak{F}(E)$ , there is an $I\in\mathcal{I}_{M_0}\cap\mathcal{I}_{M_1}$ and a partition $E=E_0\sqcup E_1$ such that $I\cap E_i$ spans $E_i$ in $M_i$ for $i\in\{0,1\}$ . This is an infinite-matroid analogue of matroid intersection, and the paper discusses it as an application of its main Cantor–Bernstein-type results. Its resolution is not established in the supplied text.

References

Primary source

Attila Joó, “A Cantor-Bernstein theorem for infinite matroids”, arXiv:2009.08439 (2022).

Additional references

4 papers in this index state this conjecture (2011–2020). The statement above is taken from the most recent of them; the others are arXiv:1404.6067, arXiv:1202.3409, arXiv:1111.0606.

infinite-matroids matroid-intersection

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 preprint claims to settle the conjecture for all infinite matroids, but the claimed proof has not been independently verified.

Nash-Williams proposed the conjecture in 1990, first for finitary matroids and later for general matroids. It asks for a common independent set whose parts span a partition of the ground set in the respective matroids.

Known results

Countable finitary matroids: proved in 2019, giving a positive answer in that case.
One matroid nearly finitary and the other dual-nearly finitary: proved by Bowler and collaborators in 2011.
Pairs satisfying the Almost Intersection Property: proved sufficient; this includes finite-rank, patchwork, partition, and certain singular cases.
The conjecture for finitary matroids implies the infinite Menger theorem.

2026 claimed proof

A 2026 arXiv preprint claims a proof for arbitrary infinite matroids, using closure-operator lemmas to obtain the required common independent set and partition. The supplied record gives no peer review or independent verification, and reports no counterexample or published correction.

Current status (as of September 2026): the conjecture remains unverified in full generality; a 2026 preprint claims a proof, while countable and several structural cases are established.

Sources

No solutions have been posted yet.