# P3098: Shelah's eventual categoricity conjecture for AECs

- ID: `P3098`
- Reference: `eventual-categoricity-aecs`
- Page: https://theoremdb.org/statements/P3098
- Record maturity: Reviewed problem with recorded work

## 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 \(\lambda'\ge H\)?

### Context

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.

### Problem setup

- **Definition (abstract elementary class).** A class of structures with a coherent strong-substructure relation satisfying directed-union and Löwenheim-Skolem axioms.
- **Definition (categorical in λ).** Exactly one model of cardinality λ up to isomorphism.
- **Remark.** 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.

### What 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.

## 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. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions.

The exact unresolved remainder is: The general ZFC theorem for arbitrary AECs remains open.

A complete resolution must meet the following acceptance conditions:
- 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.

### Background and intake notes

- 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.
- The release review checked 2 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Unresolved remainder: The general ZFC theorem for arbitrary AECs remains open. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): The general ZFC theorem for arbitrary AECs remains open.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `eventual-categoricity-aecs`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://shelah.logic.at/papers/842/
   - Also cited at S. Shelah and S. Vasey, Categoricity and multidimensional diagrams, Sh:842 (2024). eventual categoricity theorem under strongly compact cardinals
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Proves the conjectural behavior assuming a proper class of 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.
2. <a id="reference-2"></a>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 https://arxiv.org/abs/1506.07024
   - preprint; primary source; arXiv:1506.07024, checked 2026-08-01; checked 2026-08-01
   - Source 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.