We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and P -minimal theories.Our approach to Theorem 1.1, via definable types, was partly inspired by the use of Puiseux series in [11,81].Let ACVF denote the theory of (non-trivially) valued algebraically closed fields, in the ring language expanded by a predicate for the valuation divisibility. This has completions ACVF (0,0) (for residue characteristic 0), ACVF (0,p) (field characteristic 0, residue characteristic p), and ACVF (p,p) (field characteristic p). Because ACVF (0,0) is interpretable in RCVF, our methods give (non-optimal) density bounds for ACVF (0,0) (Corollary 6.3). However, they give no information on density in the theories ACVF (0,p) and ACVF (p,p) . The problems arise essentially because a definable set in 1-space in ACVF is a finite union of 'Swiss cheeses' but we have no way of choosing a particular Swiss cheese. This means that the definable types technique in our main tool (Theorem 5.7) breaks down. On the other hand, our methods do yield:Theorem 1.2. Suppose M = Q p is the field of p-adic numbers, construed as a firstorder structure in Macintyre's language L p . Then the VC density of every L p -formula ϕ(x; y) is at most 2|y| − 1.

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...

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Abstract. Let M = M, +, <, 0, {λ} λ∈D be an ordered vector space over an ordered division ring D, and G = G, ⊕, e G an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to t-topology, then it is definably isomorphic to a 'definable quotient group' U/L, for some convex -definable subgroup U of M n , + and a lattice L of rank n. As two consequences, we derive Pillay's conjecture for M as above and we show that the o-minimal fundamental group of G is isomorphic to L.

Abstract. Let R be an o-minimal expansion of a divisible ordered abelian group (R, <, +, 0, 1) with a distinguished positive element 1. Then the following dichotomy holds: Either there is a 0-definable binary operation · such that (R, <, +, ·, 0, 1) is an ordered real closed field; or, for every definable function f : R → R there exists a 0-definable λ ∈ {0} ∪ Aut(R, +) withThis has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure M := (M, <, . . . ) there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) M-definable groups with underlying set M .R. Poston showed in [8] that given an o-minimal expansion R of (R, <, +), if multiplication is not definable in R, then for every definable function f : R → R there exist r, c ∈ R such that lim x→+∞ [f (x) − rx] = c. In this paper, this fact is generalized appropriately for o-minimal expansions of arbitrary ordered groups.We say that an expansion (G, <, * , . . . ) of an ordered group (G, <, * ) is linearly bounded (with respect to * ) if for each definable function f : G → G there exists a definable λ ∈ End(G, * ) such that ultimately |f (x)| ≤ λ(x). (Here and throughout, ultimately abbreviates "for all sufficiently large positive arguments".)We now list the main results of this paper. Let R := (R, <, . . . ) be o-minimal. Theorem A (Growth Dichotomy). Suppose that R is an expansion of an ordered group (R, <, +). Then exactly one of the following holds: (a) R is linearly bounded; (b)R defines a binary operation · such that (R, <, +, ·) is an ordered real closed field. If R is linearly bounded, then for every definable f : R → R there exist c ∈ R and a definable λ ∈ {0} ∪ Aut(R, +) with Theorem B.Suppose that R is a linearly bounded expansion of an ordered group (R, <, +, 0, 1) with 1 > 0. Then every definable endomorphism of (R, +) is 0-definable. If R (with underlying set R ) is elementarily equivalent to R, then the ordered division ring of all R -definable endomorphisms of (R , +) is canonically isomorphic to the ordered division ring of all R-definable endomorphisms of (R, +).The growth dichotomy imposes some surprising constraints on continuous definable groups with underlying set R. (Here and throughout, all topological notions are taken with respect to the product topologies induced by the order topology.)