Geodesic
Regularity lemma for distal structures
Journal of the European Mathematical Society, Tome 20 (2018) no. 10, pp. 2437-2466 Cet article a éte moissonné depuis la source EMS Press

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It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and can be decomposed into very homogeneous semialgebraic pieces up to a small error (see e.g. [33, 2, 16, 18]). We show that similar results can be obtained for families of graphs with the edge relation uniformly definable in a structure satisfying a certain model-theoretic property called distality, with respect to a large class of generically stable measures. Moreover, distality characterizes these strong regularity properties. This applies in particular to graphs definable in arbitrary o-minimal structures and in p-adics.
DOI : 10.4171/jems/816
Classification : 03-XX, 05-XX, 14-XX
Keywords: NIP, VC-dimension, distal theories, o-minimality, p-adics, Erdős–Hajnal conjecture, regularity lemma 
@article{JEMS_2018_20_10_a3,
     author = {Artem Chernikov and Sergei Starchenko},
     title = {Regularity lemma for distal structures},
     journal = {Journal of the European Mathematical Society},
     pages = {2437--2466},
     year = {2018},
     volume = {20},
     number = {10},
     doi = {10.4171/jems/816},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/816/}
}
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Artem Chernikov; Sergei Starchenko. Regularity lemma for distal structures. Journal of the European Mathematical Society, Tome 20 (2018) no. 10, pp. 2437-2466. doi: 10.4171/jems/816

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