标题： 超图的平均值和高阶稳定性 显示英文标题

标题： Averages of hypergraphs and higher arity stability

摘要： 我们证明了给出概率空间中多参数集族交集测度的$k$元函数，例如$(x,y,z) \in X \times Y \times Z \mapsto \mu(P_{x,y} \cap Q_{x,z} \cap R_{y,z})$，满足一种特别强的超图正则性形式。 更一般地，这适用于较小元数函数的连续组合的（积分）平均值。 这个结果与模型论中的高元稳定性有关，我们在论文的第二部分进行讨论。 我们证明了所有同时嵌入到半单形和$GS(\mathbb{F}_3)$中的超图——这两个已知的三元稳定性失败来源——都满足类似的正则性引理——因此，不能仅通过排除的超图来描述强三元稳定性。 显示英文摘要

摘要： We show that $k$-ary functions giving the measure of the intersection of multi-parametric families of sets in probability spaces, e.g. $(x,y,z) \in X \times Y \times Z \mapsto \mu(P_{x,y} \cap Q_{x,z} \cap R_{y,z})$, satisfy a particularly strong form of hypergraph regularity. More generally, this applies to the (integral) averages of continuous combinations of functions of smaller arity. This result is connected to higher arity stability in model theory, that we discuss in the second part of the paper. We demonstrate that all hypergraphs embedding both into the half-simplex and into $GS(\mathbb{F}_3)$, the two known sources of failure of ternary stability, do satisfy an analogous regularity lemma -- hence strong ternary stability cannot be characterized simply by excluded hypergraphs.