1 · The essentials
The manuscript gives a negative answer to two general questions about organizing infinite collections under two sets of rules. It constructs two mathematical structures on the same collection, whose elements can be listed in an infinite sequence. Neither of the two proposed ways of reconciling their rules is possible for this pair. This supplies counterexamples to the unrestricted infinite matroid intersection and packing/covering conjectures. The same construction also answers a question of Joó negatively. The main result has a Lean formalization. The examples fall outside the more restricted setting of Nash-Williams’ original question, so that question is not settled by this result.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine an endless list of items governed by two organizing rulebooks. The construction supplies a pair of rulebooks for which neither of two proposed reconciliation plans works; it does not say that reconciliation fails for every more restricted kind of rulebook.
Why it matters
The counterexamples establish that the unrestricted infinite intersection and packing/covering conjectures fail in ZFC. Their significance also lies in their scope: failure already occurs for two self-dual partitional matroids on a countably infinite set. At the same time, their exclusion from the finitary setting leaves Nash-Williams’ original conjecture outside the result.
2 · Going further
The precise result concerns two matroids on a common ground set . The manuscript constructs, in ZFC, two self-dual partitional matroids with countably infinite, such that the pair admits neither a packing/covering partition nor an intersection witness. These are the two forms of existence sought by the unrestricted conjectures, so their simultaneous absence provides counterexamples to both.
The argument described in the abstract is constructive in the sense that it supplies a particular pair with the required failures. The abstract does not give the rules defining the pair or the steps establishing those failures, so no more detailed mechanism can be extracted from the supplied text. The same examples answer Joó’s question for two partitional matroids negatively.
A crucial boundary is that the examples are neither finitary nor cofinitary. Consequently, this result does not settle Nash-Williams’ original finitary conjecture. Only the main manuscript is supplied here, so no relationship to companion manuscripts can be described. The main result has a Lean formalization; the supplied information does not specify its coverage claim by claim.
Useful notions
- Countably infinite sets
- Matroids and their ground sets
- Matroid duality
- Partitional matroids
- Packing/covering partitions and intersection witnesses
What was known before / what is new
Before
The source identifies unrestricted infinite matroid intersection and packing/covering conjectures as the targets of the construction. It also identifies a question of Joó concerning two partitional matroids. Nash-Williams’ original conjecture is described as finitary, a restriction not satisfied by these examples.
New
The main result supplies two self-dual partitional matroids on a countably infinite common ground set with neither a packing/covering partition nor an intersection witness. The construction takes place in ZFC and yields counterexamples to both unrestricted conjectures. The same examples give a negative answer to Joó’s partitional-matroid question, and the main result has a Lean formalization.
Limits and assumptions
The counterexamples are specifically two self-dual partitional matroids on a countably infinite common ground set, constructed in ZFC. They are neither finitary nor cofinitary, and Nash-Williams’ original finitary conjecture remains outside the result. The abstract does not provide the construction details or the definitions of the two missing witnesses. The main result has a Lean formalization, but the supplied text does not establish that every associated claim is separately covered.
Mini glossary
- Ground set —
- The underlying collection of elements on which a matroid is defined.
- Countably infinite —
- Infinite, but with elements that can be listed in a sequence.
- Self-dual —
- Having the same matroid structure as its dual.
- Counterexample —
- An instance satisfying a proposed statement’s assumptions but not its conclusion.
- Lean formalization —
- A representation of a mathematical result and its justification that can be checked in Lean.
Keywords : Infinite matroid intersection · Packing/covering · Counterexamples
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A Counterexample to the Infinite Matroid Packing/Covering Conjecture
24/09/2026We construct in ZFC two self-dual partitional matroids on a countably infinite common ground set that admit neither a packing/covering partition nor an intersection witness. This disproves the unrestricted infinite matroid packing/covering and intersection conjectures and answers Joó's question for two partitional matroids negatively. The examples are neither finitary nor cofinitary.