The aim of this note is to overview some of our work in Chernikov, Towsner'20 (arXiv:2010.00726) developing higher arity VC theory (VC$_n$ dimension), including a generalization of Haussler packing lemma, and an associated tame (slice-wise) hypergraph regularity lemma; and to demonstrate that it characterizes higher arity PAC learning (PAC$_n$ learning) in $n$-fold product spaces with respect to product measures introduced by Kobayashi, Kuriyama and Takeuchi'15. We also point out how some of the recent results in arXiv:2402.14294, arXiv:2505.15688, arXiv:2509.20404 follow from our work in arXiv:2010.00726.

翻译：本文旨在概述我们在Chernikov与Towsner'20（arXiv:2010.00726）中发展高阶元VC理论（VC$_n$维）的部分工作，包括Haussler打包引理的一个推广，以及一个相关的驯顺（分层）超图正则引理；并证明该理论刻画了在Kobayashi、Kuriyama与Takeuchi'15引入的乘积测度下，$n$重乘积空间中的高阶元PAC学习（PAC$_n$学习）。我们还指出，arXiv:2402.14294、arXiv:2505.15688、arXiv:2509.20404中的一些近期结果如何从我们在arXiv:2010.00726的工作中推导得出。