Result 185, Combinatorics

Counterexamples to infinite matroid intersection and packing/covering

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Matroids describe which selections from a set count as independent, abstracting ideas from graphs and linear algebra. The manuscript claims that two such systems on an infinite set can defeat conjectured ways of coordinating their independence.

What changes?

The manuscript reports two partitional matroids, built from separate blocks, on one countably infinite set. Both are self-dual: each is equivalent to its dual, obtained by replacing maximal independent sets with their complements. Constructed in ZFC, the standard axioms of set theory, they admit neither a packing/covering partition nor an intersection witness, the structures demanded by the respective conjectures. This disproves both unrestricted conjectures and answers Joó's question for two partitional matroids negatively. The examples are neither finitary nor cofinitary.

What does that help mathematicians do?

The examples show that block structure and self-duality alone cannot guarantee the desired intersection or packing/covering structures. Researchers pursuing valid infinite extensions therefore need additional restrictions. Nash-Williams' original finitary conjecture remains untouched: in finitary matroids, every dependent set contains a finite dependent subset. These counterexamples fall outside that class and its dual class, so they do not settle those more restricted settings.

Are there practical applications?

Its immediate value is foundational: the construction supplies test cases for proposed infinite-matroid theorems. Because it works in ZFC, the obstruction does not require extra set-theoretic assumptions. This is a nonexistence result, not an algorithm for finding intersections, packings or coverings; its contribution is to clarify which universal guarantees researchers cannot rely on.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A Counterexample to the Infinite Matroid Packing/Covering Conjecture

September 24, 2026 20 pages Main result formalized in Lean

We 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.

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Cite (BibTeX) 
@misc{OAI:A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026,
  author = {{OpenAI}},
  title = {{A Counterexample to the Infinite Matroid Packing/Covering Conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf}{OAI:A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/185.md.

Counterexamples to infinite matroid intersection and packing/covering

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The infinite matroid packing/covering conjecture predicts a partition of a common ground set into parts admitting the corresponding packing and covering. The formalization constructs two self-dual partitional matroids on one countably infinite ground set that admit neither an independent covering nor a packing/covering partition. The same pair has no intersection witness, so it also refutes the unrestricted infinite matroid intersection conjecture.

Separate formalized consequences give counterexamples to the individual covering and packing conjectures. The construction is in ordinary set theory with Choice and assumes no finitary restriction on the matroids.

Comparator links

Result Comparator statement
Infinite matroid packing/covering counterexample InfiniteMatroid.lean
Partitional intersection, covering, and packing counterexamples InfiniteMatroidCorollaries.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.