We investigate various forms of (model-theoretic) stability for hypergraphs and their corresponding strengthenings of the hypergraph regularity lemma with respect to partitions of vertices. On the one hand, we provide a complete classification of the various possibilities in the ternary case. On the other hand, we provide an example of a family of slice-wise stable 3-hypergraphs so that for no partition of the vertices, any triple of parts has density close to 0 or 1. In particular, this addresses some questions and conjectures of Terry and Wolf. We work in the general measure theoretic context of graded probability spaces, so all our results apply both to measures in ultraproducts of finite graphs, leading to the aforementioned combinatorial applications, and to commuting definable Keisler measures, leading to applications in model theory.

翻译：我们研究了超图的（模型论）稳定性在不同形式下的表现，以及相应的顶点划分下超图正则引理的加强版本。一方面，我们在三元情况下提供了各种可能性的完整分类。另一方面，我们构造了一个切片稳定3-超图族，使得对于任何顶点划分，任意三个部分之间的密度既不接近0也不接近1。具体而言，这解决了Terry和Wolf提出的若干问题与猜想。我们在分级概率空间的一般测度论框架下进行研究，因此所有结果既适用于有限图超积中的测度（从而导出前述组合应用），也适用于可交换的可定义Keisler测度（从而导出模型论中的应用）。