Kalai's conjecture for tight trees
Kalai proposed in 1984, in a form recorded by Frankl and Füredi, a hypergraph generalisation of the Erdős-Sós conjecture: for a suitable notion of a tight tree with edges in an -uniform hypergraph, every -graph on vertices with more than edges should contain . The case is the Erdős-Sós conjecture, proved in 2026. Does the conjecture hold for all ?
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Extremal hypergraph theory
- Posed by
- Gil Kalai (1984), recorded by Péter Frankl and Zoltán Füredi; the $r=2$ case is the Erdős-Sós conjecture
- Year posed
- 1984
- Years open
- 42y
- Solved
- 2026-09-07
- Model
- GPT-6 Astra
- Vendor
- OpenAI
- Collaborators
- Dhruv Mubayi, Jacques Verstraëte
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
From the paper: the proof was found by GPT-6 Astra, extending its own method of proof for the Erdős-Sós conjecture to the hypergraph setting. The authors supplied the framing and the write-up.
Verification
Checked here on 22 September 2026 against arXiv:2609.08012: the introduction states that Kalai proposed the hypergraph generalisation in 1984, recorded by Frankl and Füredi, that the case is the Erdős-Sós conjecture, and that the conjecture is proved here via a shadow bound whose equivalence to Kalai's conjecture was established by Füredi, Jiang, Kostochka and the authors. The mathematics was not checked here; fifteen days old, no referee. Same authors and same model as the digraph entry three days later.