The Abbott-Hanson Recurrence for Schur Numbers
The classical Abbott-Hanson recurrence gives for Schur numbers, and had stood as the basis for the best asymptotic lower bounds. Shifted -templates, a more flexible form of Rowley's template construction, yield and hence improved lower bounds.
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Ramsey theory
- Posed by
- Harvey Abbott, Denis Hanson
- Year posed
- 1972
- Years open
- 54y
- Solved
- 2026-07-16
- Model
- ChatGPT 5.5 Pro
- Vendor
- OpenAI
- Collaborators
- Nils Bengone, Amine Brouk, Max Grinsztajn, Terence Helbert, Bao Lugherini, Arpad Rimmel, Joanna Tomasik
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
improves the classical recurrence; the exact Schur numbers beyond S(5) remain unknown
What the AI did
The abstract says the shifted -template construction was discovered during a conversation with ChatGPT 5.5 Pro and then refined, verified and extended to multiple templates by the authors, and the conclusion describes the paper as building on an original idea of the model.
Verification
arXiv preprint; not yet peer-reviewed.
Source
arXiv:2607.15034 - Shifted S-templates and improved lower bounds for Schur numbers