OpenAI · 185-01 · 2026-09-24 · Combinatorics
A Counterexample to the Infinite Matroid Packing/Covering Conjecture
preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026
Family 185. Counterexamples to infinite matroid intersection and packing/covering.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 185: Counterexamples to infinite matroid intersection and packing/covering. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.