Rota's 1970 unimodality conjecture for matroid flats disproved, with ChatGPT 5.6 Pro suggesting a key construction
In July–August 2026 three arXiv papers overturned classic matroid conjectures with AI help. Matt Larson (2607.02208, Jul 2) disproved Mason's log-concavity conjecture for flat counts and White's exchange conjecture, using ChatGPT 5.5 Pro (and Claude Opus 4.8) to search for counterexamples. Divoux, Lowen and Wang (2607.22515, Jul 24) then disproved Rota's 1970 conjecture that the numbers of flats by rank form a unimodal sequence; ChatGPT 5.6 Pro supplied a q-lift example they generalized. A follow-up (2608.07342, Aug 7) showed the sequence can have arbitrarily many peaks.
Key facts
- Rota's conjecture (1970): the Whitney numbers of the second kind (number of flats of each rank) of every matroid form a unimodal sequence
- arXiv 2607.02208 (Matt Larson, Jul 2, 2026): counterexamples to Mason's conjecture (log-concavity of flat counts; e.g. W_74^2 < W_73·W_75 for a graphic matroid of a generalized theta graph with three 26-edge paths and one edge) and to White's conjecture on symmetric exchanges (a rank-9 binary matroid on 18 elements)
- Larson's AI use: 'I prompted ChatGPT 5.5 Pro to look for a counterexample around 20 times'; the paper also credits Claude Opus 4.8
- arXiv 2607.22515 (Alexander Divoux, Chayim Lowen, Shouda Wang, Jul 24, 2026; 8 pages): 'We give counterexamples to Rota's 1970 conjecture'; built from Larson's matroid via Whittle's q-lift
- AI role in 2607.22515: 'We used ChatGPT 5.6 Pro to search for a counterexample to this weaker property. Based on a suggestion of the authors, ChatGPT 5.6 Pro returned a non-convex example built using a q-lift'; the paper was 'written entirely by the authors'
- arXiv 2608.07342 (Divoux, Larson, Lowen, Wang, Aug 7, 2026): flat counts can have arbitrarily many peaks
- Listed in Table 3 of the AI4Math survey arXiv 2608.24961 as a conjecture disproved with ChatGPT 5.6 Pro
Science result
- Field
- mathematics / combinatorics (matroid theory)
- Problem
- Rota's unimodality conjecture for the number of flats of a matroid by rank (and Mason's log-concavity conjecture) (open since 1970)
- Result
- Explicit matroids whose flat counts are not unimodal (Rota), not log-concave (Mason), and can have arbitrarily many peaks.
- AI system
- ChatGPT 5.6 Pro, ChatGPT 5.5 Pro, Claude Opus 4.8
- Human role
- AI-assisted: humans directed the search and wrote the papers; ChatGPT found initial counterexamples/constructions that the authors generalized
- Verification
- Explicit, checkable counterexamples in arXiv preprints; not yet peer-reviewed
- Status
- confirmed
- Why surprising
- A 56-year-old conjecture in a mainstream area of combinatorics fell in a few weeks once an AI-found counterexample to a stronger conjecture appeared.
What happened
Larson first used ChatGPT 5.5 Pro, prompted around 20 times with hints (for example, look at matroids realizable over F5 and at failures at high indices), to find a graphic matroid that breaks Mason's log-concavity conjecture. Divoux, Lowen and Wang then asked ChatGPT 5.6 Pro to look for an example of a weaker property; it returned a q-lift construction, which the authors turned into counterexamples to Rota's unimodality conjecture. All four then showed the flat-count sequence can have many peaks.
Why it matters
Rota's conjecture was one of the best-known open problems about matroids. The chain is a clear example of the 2026 pattern: AI systems search for counterexamples, and humans generalize and write up the results, with the AI's role stated precisely in the papers.
Changelog
- 2026-09-30: created (resolves the leads.md line on Rota's conjecture for flats)
Related events
- 'The Gold Rush in AI4Math': substantive AI use in arXiv math papers rises from 1.4% to 14% in five months ★★★
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
Sources (4)
- paperarXiv 2607.22515: Matroid flat counts are not unimodal
- paperarXiv 2607.02208: Counterexamples to two conjectures about matroids (Larson)
- paperarXiv 2608.07342: Matroid flat counts can have many peaks
- paperarXiv 2608.24961: The Gold Rush in AI4Math (Table 3)
id: 2026-07-24-rota-flats-unimodality-disproved · updated 2026-09-30 · open in the interactive timeline