Submission date: 26 July 2016

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. Lovász, B. Szegedy, “Regularity partitions and the topology of graphons”, An irregular mind, Springer Berlin Heidelberg, 2010, 415-446]. Besides, we revise the two special cases of regularity for stable and distal hypergraphs, improving and generalizing the results from [A. Chernikov, S. Starchenko, “Regularity lemma for distal structures”, Journal of the European Mathematical Society, to appear, arXiv:1507.01482] 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.

Mathematics Subject Classification: 03C45, 05C35, 05C25

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Full text arXiv 1607.07701: pdf, ps.