Definable regularity lemmas for NIP hypergraphs

TL;DR

This paper develops a model-theoretic framework for regularity lemmas in NIP hypergraphs, extending classical results and exploring homogeneous definable subsets with new positive results and counterexamples.

Contribution

It introduces a systematic study of regularity phenomena for NIP hypergraphs, connecting to stable measures and generalizing existing regularity lemmas for stable and distal hypergraphs.

Findings

Established a hypergraph version of regularity lemmas for NIP structures.

Improved and generalized regularity results for stable and distal hypergraphs.

Provided new results and counterexamples on the existence of large homogeneous definable subsets.

Abstract

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 from [L. Lov\'asz, B. Szegedy, "Regularity partitions and the topology of graphons", An irregular mind, Springer Berlin Heidelberg, 2010, 415-446]. Besides, we revise the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the results from [A. Chernikov, S. Starchenko, "Regularity lemma for distal structures", J. Eur. Math. Soc. 20 (2018), 2437-2466] and [M. Malliaris, S. Shelah, "Regularity lemmas for stable graphs", Transactions of the American Mathematical Society, 366.3, 2014, 1551-1585]. 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.