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.

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

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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.

中文翻译：

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抽象基本类中的类似于简单的独立关系

我们使用怪兽模型在AEC的背景下介绍和研究简单和超简单的独立关系。

定理0.1

令 K 为具有怪兽模型的AEC。

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如果 K 具有简单的独立关系，则 K 不具有2树属性。

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如果 K 与 $（<{\mathrm{\aleph }}_0）$-witness属性用于单例，则 K 不具有tree属性。

这两个事实的证明是通过在固定模型上找到与大型子集不一致的小型Galois型类别的基本界限来完成的。我们认为这种更好的类型计数方法本身就是一个有趣的概念。

在独立性关系的局部性假设下，我们通过Lascar等级的有限性来描述超简单的独立性关系。