AI 中文总结
研究在NIP强k-远结构中可定义的(k + 1)-一致超图，通过引入k-强诚实定义，证明其满足齐次正则引理，扩展了Chernikov和Starchenko的相关成果，将强诚实定义理论推广到高元情形。
AI 中文摘要
我们证明了在NIP强k-远结构中可定义的(k + 1)-一致超图满足齐次正则引理，即它们可被划分为有限个单纯复形，其中大部分是齐次的。此外，划分的部分可以统一可定义地选取，划分的大小是误差参数倒数的多项式。这扩展了Chernikov和Starchenko为远结构中可定义的超图证明的齐次正则引理。我们通过引入k-强诚实定义并证明NIP结构是强k-远的当且仅当每个公式都有k-强诚实定义来证明此结论，这将远结构中的强诚实定义理论扩展到了高元情形。
英文摘要
We prove that $(k+1)$-uniform hypergraphs definable in an NIP strongly $k$-distal structure satisfy a homogeneous regularity lemma -- they can be partitioned into a bounded number of simplicial complexes, most of which are homogeneous (meaning that the restriction of the hypergraph to the simplicial complex is either complete or empty). Furthermore, the parts of the partition can be chosen uniformly definably, and the size of the partition is polynomial in the reciprocal of the error parameter. This extends the homogeneous regularity lemma proven by Chernikov and Starchenko for hypergraphs definable in a distal structure. We prove this by introducing $k$-strong honest definitions and showing that an NIP structure is strongly $k$-distal if and only if every formula $φ(x_1, ..., x_k; y)$ has a $k$-strong honest definition. This extends the theory of strong honest definitions in distal structures to the higher-arity setting.
Comments26 pages
