Can an infinite self-dual uniform matroid be constructed in ZFC?
An infinite matroid is uniform if exchanging any element of a basis for any element outside it again gives a basis. Bowler and Geschke developed a basis criterion for infinite uniform matroids and constructed countable self-dual uniform examples under Martin's axiom for countable posets, hence under the continuum hypothesis, and left open whether such a matroid exists in ZFC alone. Gollin and Joo recorded even the ZFC existence of a uniform matroid of infinite rank and infinite corank as open. Is there, provably in ZFC, an infinite self-dual uniform matroid?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Infinite matroid theory; set theory
- Posed by
- Nathan Bowler and Stefan Geschke (Proc. AMS 2016); recorded again by Gollin and Joo (2025)
- Year posed
- 2016
- Years open
- 10y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 12 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
The manuscript constructs, in ZFC, a self-dual uniform matroid on a countable set with every basis and its complement infinite, answering the Bowler-Geschke question and the Gollin-Joo existence question affirmatively. Its bases satisfy an extra ordinal inequality used to build the packing/covering counterexample. It does not address uncountable ground sets or classify such matroids.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. This answer is a building block of the infinite matroid packing/covering counterexample in the same manuscript.
Verification
No independent mathematician has checked this yet. Checked here: the introduction's claim and Remark (uniform local matroid) were read against the question as the paper quotes it from Bowler-Geschke and Gollin-Joo. The Lean challenge InfiniteMatroidCorollaries.json (theorem OAI.InfiniteMatroidCounterexample.partitional_intersection_counterexample, module OAI.Combinatorics.InfiniteMatroid.Corollaries, file exists at the pinned commit, not in the formalization catalogue) states the existence of a self-dual uniform Mathlib matroid Q on an infinite countable type. Left unreviewed under the strict tier rule: the question is about provability in ZFC, and the Lean statement expresses existence in Lean's own foundations, not ZFC, so it does not state the headline as posed. Not rebuilt here.