Regularity lemma for distal structures

TL;DR

This paper extends regularity lemmas from semialgebraic graphs to those definable in distal structures, characterizing strong regularity properties via model-theoretic distality, with applications to o-minimal structures and p-adics.

Contribution

It establishes that graphs definable in distal structures exhibit strong regularity properties and characterizes these properties through distality, broadening the scope beyond semialgebraic graphs.

Findings

Graphs definable in distal structures satisfy strong regularity properties.

Distality characterizes the regularity properties of such graphs.

Applications include graphs in o-minimal structures and p-adic fields.

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.