Regularity lemma for distal structures

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.1 2 ARTEM CHERNIKOV AND SERGEI STARCHENKO Theorem 1.2 (Erdős, Hajnal and Pach [14]). For any finite graph H there is a constant δ = δ(H) > 0 such that every H-free graph on n vertices has a homogeneous pairThe following definition is taken from [17].Remark 1.4. Is is shown in [2] that if a family of finite graphs G has the strong Erdős-Hajnal property and is closed under taking induced subgraphs then it has the Erdős-Hajnal property.In this paper we consider families of graphs whose edge relations are given by a fixed definable relation in a first-order structure.Definition 1.5. Let M be a first-order structure and R ⊆ M k × M k be a definable relation. Consider the family G R of all finite graphs V = (G, E) where G ⊆ M k is a finite subset and E = (V ×V )∩R. We say that R satisfies the (strong) Erdős-Hajnal property if the family G R does.We extend this notion to the bi-partite case. Definition 1.6. Let M be a first-order structure and R ⊆ M m × M n a definable relation. (1) A pair of subsets A ⊆ M m , B ⊆ M n is called R-homogeneous if either A × B ⊆ R or (A × B) ∩ R = ∅. (2) We say that the relation R satisfies the strong Erdős-Hajnal property if there is a constant δ = δ(R) > 0 such that for any finite subsets A ⊆ M m , B ⊆ M n there are A 0 ⊆ A, B 0 ⊆ B with |A 0 | ≥ δ|A|, |B 0 | ≥ δ|B|, and the pair A 0 , B 0 is R-homogeneous.Our motivation for this work comes from the following remarkable theorem by Alon et al.Theorem 1.7 ([2, Theorem 1.1]). If R ⊆ R n × R m is a semialgebraic relation then R has the strong Erdős-Hajnal property.Remark 1.8.(i) Although it is not stated explicitly in [2], but can be easily derived from the proof, homogeneous pairs in the above theorem can be chosen to be relatively uniformly definable. (ii) The above theorem was generalized by Basu (see [5]) to (topologically closed) relations definable in arbitrary o-minimal expansions of real closed fields.Besides the Erdős-Hajnal property for semialgebraic graphs, the above theorem has many other applications including unit distance problems [32], improved bounds in higher dimensional semialgebraic Ramsey theorem [11], [2, Theorem 1.2], algorithmic property testing [18], and can also be used to obtain a strong Szemerédi-type regularity lemma for semialgebraic graphs [16,18] (see also Section 5).The aim of this article is to demonstrate that the above r...

Wild theories with o-minimal open core

“…Although we do not know it, we expect all examples of non-distal NIP expansions of (R, <, +) produced in [13] to have distal expansions. So far the only known NIP theory without an distal expansion is the theory of algebraically closed fields of characteristic p by [4,Proposition 6.2]. Combining this with Theorem B, we almost immediately obtain the following.…”

“…It is worth pointing out in this section on strong dependence that by Dolich and Goodrick [5, Corollary 2.4] every strongly dependent expansion of the real field has o-minimal open core. In contrast to this restriction, our Theorem B (4) shows that there is a large variety of such expansions of the real field.…”

“…Statements (3) and (4) of Theorem A indicate that we can choose T * in such a way that not only the open core of T * is o-minimal, but also T * remains tame in the sense of Shelah's combinatorial tameness notions. For definitions of NIP and strong dependence, we refer the reader to Simon [19].…”

Stable arithmetic regularity in the finite field model

“…Fix τ ∈ 2 t+1 , so τ = η ∧ i for some i ∈ {0, 1} and η ∈ 2 t . Observe that (v) holds for τ by definition of X η∧i , (iii) holds for τ by (6), and (ii) holds for τ by our choice of g η∧i . For (iv), note that…”

Definable sets containing productsets in expansions of groups

“…Finally, we consider the finitary productset property in the setting of distal groups. Using recent work of Chernikov and Starchenko [2], we show that for distal groups, the finitary productset property is always witnessed by a definable family of sets. Using this, we show that if G is a distal expansion of an amenable group, and if G eliminates the quantifier ∃ ∞ , then any definable subset of G with positive Banach density has the productset property witnessed by definable sets.…”