Regularity lemma for distal structures

Artem Chernikov; Sergei Starchenko

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arXiv · 1507.01482

Regularity lemma for distal structures

Artem Chernikov·Sergei Starchenko

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Abstract

It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into very homogeneous semialgebraic pieces up to a small error (e.g., see [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.

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Artem Chernikov, Sergei Starchenko. 2015-07-06. Regularity lemma for distal structures. https://arxiv.org/abs/1507.01482

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