Submission date: 5 March 2020

Abstract:

We introduce and study *-simple, simple and supersimple independence relations in the context of AECs with a monster model.
Theorem
Let K be an AEC with a monster model. - If K has a *-simple independence relation, then the relation is canonical, K is stable and K does not have the tree property.
- If K has a simple independence relation with (< ℵ_0)-witness property, then K does not have the tree property.
The proof of both facts is done by finding cardinal bounds to classes of small Galois-types over a fixed model that are inconsistent for large subsets. We think this finer way of counting types is an interesting notion in itself.
We characterize supersimple independence relations by finiteness of the Lascar rank under locality assumptions on the independence relation.

Mathematics Subject Classification: 03C48, 03C45, 03C55

Keywords and phrases:

Full text arXiv 2003.02705: pdf, ps.