Rota's Unimodality Conjecture for Matroid Flats
Is the sequence counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Matroid theory
- Posed by
- Gian-Carlo Rota
- Year posed
- 1970
- Years open
- 56y
- Solved
- 2026-07-24
- Model
- ChatGPT-5.6 Pro
- Vendor
- OpenAI
- Collaborators
- Alexander Divoux, Chayim Lowen, Shouda Wang
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 36 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
A counterexample to the convexity of the sequence was found with ChatGPT-5.6 Pro following a suggestion of the authors. This matroid was constructed using Whittle's -lift mechanism, which the authors then adapted into a counterexample for unimodality.
Verification
Public arXiv preprint with explicit counterexamples, checked by the human authors. Not yet peer-reviewed.