Counterexamples to infinite matroid intersection and packing/covering

The explanation

Take four letters, a, b, c and d. Call a set of letters independent if it has at most two letters. A largest independent set is called a basis. Here every pair is a basis, and there are six of them. This rule is a small example of a matroid.

A matroid is a ground set plus a rule saying which subsets are independent. The rule must behave like independence of arrows in space. Every matroid has a dual, whose bases are the complements of the original bases. In the four-letter example, the complement of a pair is another pair. So the dual is the same matroid, and the matroid is called self-dual.

Now use the four-letter matroid twice and ask a covering question. Can one independent set from each copy together contain every letter? Yes: take {a, b} from the first copy and {c, d} from the second. For a self-dual matroid used twice, a basis and its complement always work this way.

The ground set can also be infinite. Infinite matroids need extra care, and the standard axioms come from Bruhn, Diestel, Kriesell, Pendavingh and Wollan. A covering of a family of matroids picks one independent set per matroid, and together they contain the whole ground setSource 1. A packing picks disjoint spanning sets instead. Bowler and Carmesin conjectured that every family splits the ground set into a part with a packing and a part with a coveringSource 2. A related conjecture, matroid intersection, predicts a common independent set with a matching property.

This paper builds two self-dual matroids on one countably infinite ground set, meaning one that can be listed as first, second, third and so on. No independent set from the first and one from the second ever cover the ground set. The pair has no packing/covering split, so the conjecture is falseSource 3. The paper also deduces that the unrestricted intersection conjecture is falseSource 4. The same pair answers a question of Attila Joó negativelySource 5. All of this uses only ZFC, the standard axioms of set theory with the axiom of choice.

The main obstacle was building the basic piece. Bowler and Geschke had built countable self-dual examples, but only by assuming an extra axiomSource 6. Whether one exists using ZFC alone was left openSource 7. The paper builds such a matroid, called Q, on a countable set in ZFCSource 8. The proof gives certain large sets a rank measured by ordinals. Ordinals are numbers with one key property: no chain of them can decrease forever. The paper places copies of Q along a path that runs forever in both directions. A covering would force the rank to drop at every step, which cannot happenSource 9.

The examples are unusual in one way that limits what they settle. A matroid is finitary when a set is independent as soon as all its finite parts are. These examples are neither finitary nor have finitary dualsSource 10. The original conjecture of Nash-Williams concerned finitary matroids. This paper does not refute that original formSource 11.

How sure is this? A computer proof checker called Lean has checked formal statements of the main results. The Lean scope note says they cover the two self-dual matroids with no packing/covering split and no intersection witness. They also cover the separate covering and packing consequences. The note says the construction assumes no finitary restriction. No specialist review of the paper has been published.

What it doesn't show. This does not refute Nash-Williams' original conjecture, which is about finitary matroids.