Petrykowski's conjecture on bounded orbits and definable amenability
For a definable group in a first-order theory, Petrykowski proposed that admitting a global type whose left-translation orbit is bounded should suffice for the group to be definably amenable; Newelski recorded the conjecture and proved it under stronger hypotheses. Does a bounded left-translation orbit imply definable amenability?
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Model theory
- Posed by
- Marcin Petrykowski; recorded as a conjecture by Ludomir Newelski, who proved it under stronger hypotheses
- Year posed
- 2012
- Years open
- 14y
- Solved
- 2026-09-04
- Model
- ChatGPT 5.6
- Vendor
- OpenAI
- Collaborators
- Artem Chernikov
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
From the paper's AI disclosure: ChatGPT 5.6 was used to establish parts of the general model-companion construction and to use finite quotients to construct the bounded orbit. The framing, the extension of the earlier seven-author construction and the write-up are the author's.
Verification
Checked here on 22 September 2026 against arXiv:2609.05711: the abstract states the counterexample as a simple expansion of the theory of nonabelian free groups with a global type of bounded left-translation orbit that is not definably amenable, extending the construction of Chernikov, Hrushovski, Kruckman, Krupiński, Moconja, Pillay and Ramsey. The introduction credits Petrykowski for the proposal and Newelski for recording it, and the reference list carries Newelski, Isr. J. Math. 187 (2012), and Newelski-Petrykowski in JLMS. The mathematics was not checked here; eighteen days old, no referee.