Simple-like independence relations in abstract elementary classes

Abstract

We introduce and study simple and supersimple independence relations in the context of AECs with a monster model.

Theorem 0.1

Let K be an AEC with a monster model.

• If K has a simple independence relation, then K does not have the 2-tree property.

• If K has a simple independence relation with the $(<{\mathrm{\aleph }}_0)$-witness property for singletons, 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 that 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.