Homogeneous hypergraph regularity lemmas via $k$-strong honest definitions

Abstract

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 $\varphi(x_1, ..., x_k; y)$ has a $k$-strong honest definition. This extends the theory of strong honest definitions in distal structures to the higher-arity setting.