[#P3098] Shelah's eventual categoricity conjecture for AECs
Problem. For every abstract elementary class \(K\) with Loewenheim-Skolem number \(\kappa\) and arbitrarily large models, is there a cardinal \(H=H(\kappa)\) such that categoricity of \(K\) in one \(\lambda\ge H\) implies categoricity in every \(\lambda'\ge H\)?
1Context
Known frontier: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Open boundary: The general ZFC theorem for arbitrary AECs remains open.
2Problem setup
Definition 1 (abstract elementary class). A class of structures with a coherent strong-substructure relation satisfying directed-union and Löwenheim-Skolem axioms.
Definition 2 (categorical in λ). Exactly one model of cardinality λ up to isomorphism.
Remark 1. Categoricity means uniqueness of the model at a given size. Morley's theorem gives an eventual transfer for first-order theories; the conjecture asks for its broad abstract-elementary analogue.
3What counts as a solution
- Prove the stated tail transfer in ZFC with an explicit or definable threshold H(κ).
- Or construct an AEC with arbitrarily large models categorical once above the proposed threshold but failing categoricity later.
1Status
Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Exact unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.[1][2]
1Records
Notes and companion material
Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Exact unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.
- Equivalent-formulation queries: Shelah eventual categoricity conjecture AEC open 2026; eventual categoricity abstract elementary classes large cardinals 2024
- Strongest checked neighboring result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions.
- Exact unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.
How the 4 records connect
ProblemShelah's eventual categoricity conjecture for AECs
2See also
How to cite
TheoremDB contributors, “Shelah's eventual categoricity conjecture for AECs,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/eventual-categoricity-aecsThis page as plain text: eventual-categoricity-aecs.md
This problem includes 4 records joined by 3 typed links, sourced from shelah.logic.at[1], current as of August 1, 2026.
1References
- Packet source. S. Shelah and S. Vasey, Categoricity and multidimensional diagrams, Sh:842 (2024). eventual categoricity theorem under strongly compact cardinals. eventual categoricity theorem under strongly compact cardinals. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves the conjectural behavior assuming a proper class of strongly compact cardinals.Also cited at S. Shelah and S. Vasey, Categoricity and multidimensional diagrams, Sh:842 (2024). eventual categoricity theorem under strongly compact cardinals.Source used to assess the problem's recorded status.For Shelah's eventual categoricity conjecture for AECs: This is the dated publication status for the canonical target Shelah's eventual categoricity conjecture for AECs.Source named by the research packet.
- Sebastien Vasey, “Shelah's eventual categoricity conjecture in universal classes: part I”. Annals of Pure and Applied Logic 168 (2017), no. 9, 1609-1642. DOI 10.1016/j.apal.2017.03.003. arXiv:1506.07024 (2015). Conjecture 1.2 and main theorem. ↗preprint · primary source · arXiv:1506.07024, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Proves substantial transfer results for universal classes and states the general AEC conjecture.Source used to assess the problem's recorded status.For Shelah's eventual categoricity conjecture for AECs: Proves substantial transfer results for universal classes and states the general AEC conjecture.
Original TheoremDB editorial statement and source synthesis; external works are used for citation only.