Kahn's flow conjecture
Chvátal conjectured in 1974 that among the intersecting subfamilies of any downset (a family closed under taking subsets), one of maximum size can be taken to be a star: all sets containing some fixed element. Kahn's flow conjecture is a strengthening, phrased as the existence of a flow between the family and a star, which implies Chvátal's statement. Is Kahn's flow conjecture true?
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Extremal set theory
- Posed by
- Jeff Kahn, as a strengthening of Vašek Chvátal's 1974 conjecture on intersecting subfamilies of a downset
- Year posed
- 1974
- Years open
- 52y
- Solved
- 2026-09-20
- Model
- GPT-6 Astra
- Vendor
- OpenAI
- Collaborators
- Peter Keevash
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The paper's statement on AI use: the proof was found by GPT-6 Astra, following an approach suggested by the author. The author supplied the route and the surrounding write-up; the model produced the argument along it.
Verification
Checked here on 22 September 2026 against arXiv:2609.23595: the abstract states that Kahn's flow conjecture is proved and describes it as a strong form of Chvátal's conjecture, and the reference list confirms the attribution chain (Chvátal 1974, Hypergraph Seminar, LNM 411; Friedgut, Kahn, Kalai and Keller, JCTA 156 (2018), on Chvátal's conjecture and correlation inequalities). The mathematics was not checked here. Two days old, no referee. A separate September 2026 preprint (arXiv:2609.19123, Chang, Liu and Liu) proves Chvátal's conjecture itself by a correlation inequality and is cited here; the two are independent.