Browse › Combinatorics › 185 Counterexamples to infinite matroid intersection and packing/covering
A Counterexample to the Infinite Matroid Packing/Covering Conjecture
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Counterexamples to infinite matroid intersection and packing/covering: Lean scope note (OpenAI math release, lean/docs/185.md, 2026-10-06). The release's Lean scope notes say the formalization constructs the two countable self-dual partitional matroids and proves they have no independent covering, packing/covering partition or intersection witness, with separate files for the universal covering, packing and partitional-intersection corollaries. [AI-generated per source] [family-level]
Possibly relevant literature (found by the agent, not checked)
- Irfan Alam, Matroid intersection and packing/covering conjectures are true in the class of finitary matroids (arXiv:2501.04321, January 2025; withdrawn in v3). This claimed proof of the finitary cases was withdrawn after Nathan Bowler pointed out a critical error, according to the arXiv comment; it concerns the finitary setting that the paper says its result does not cover.
- Nathan Bowler, A breakthrough on the matroid intersection conjecture (The Matroid Union blog, 17 September 2021). This survey of Joó's countable result lists dropping the finitary/cofinitary restrictions as an open question, which is the general setting the counterexample addresses.
- Nathan Bowler, Infinite Matroids (habilitation thesis, Universität Hamburg). The thesis has a chapter on the Packing/Covering Conjecture and a section titled "The complexity of self-dual uniform matroids", the two topics of the paper's Theorem 1.1 and its ZFC construction of the self-dual uniform matroid $Q$.
Digests and expositions
- Latent Seminar, Infinite matroid packing, covering, and intersection (2026-10-07). This exposition explains the double-ray ordinal-descent obstruction, the self-dual reduction and the distinction from finitary results, while treating the auxiliary set-theoretic construction as an input and disclaiming independent verification of the complete proofs. [family-level]
- Open Mathematics Courses, Packing and covering in infinite matroids (October 2026). Five lessons by Claude Opus 5.5, with additions by GPT-6 Astra, develop the ideals, ultrafilter rank, local bases, self-dual assembly and covering obstruction, with exercises and an explicit statement that independent review is incomplete. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 185
Independent sites; their content is not reviewed here.
Cite
@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 provides Lean files for this family: lean/docs/185.md. The Lean statements have not been checked by people against the paper's claims.
Family 185: 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.
This is the only paper in this family.