Definable Regularity Lemmas for Nip Hypergraphs

We present a systematic study of the regularity phenomena for NIP hypergraphs and connections to the theory of (locally) generically stable measures, providing a model-theoretic hypergraph version of the results of Alon-Fischer-Newman and Lovász-Szegedy for graphs of bounded VC-dimension. We also consider the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the corresponding results for graphs in the literature. Finally, we consider a related question of the existence of large (approximately) homogeneous definable subsets of NIP hypergraphs and provide some positive results and counterexamples, in particular for graphs definable in the p-adics.