Related papers: Glivenko-Cantelli classes and NIP formulas

We apply the work of Bourgain, Fremlin and Talagrand on compact subsets of the first Baire class to show new results about phi-types for phi NIP. In particular, we show that if M is a countable model, then an M-invariant phi-type is Borel…

We prove a definable version of Matou\v{s}ek's $(p,q)$-theorem in NIP theories. This answers a question of Chernikov and Simon. We also prove a uniform version. The proof builds on a proof of Boxall and Kestner who proved this theorem in…

The main purpose of this paper is to present a new and more uniform model-theoretic/combinatorial proof of the theorem ([5]): The randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$ is a (complete) first-order continuous…

In this paper, we show that a partitioned formula \phi is dependent if and only if \phi has uniform definability of types over finite partial order indiscernibles. This generalizes our result from a previous paper [1]. We show this by…

We study generically stable measures in the local, NIP context. We show that in this setting, a measure is generically stable if and only if it admits a natural finite approximation.

We prove that if \mu is a generically stable stable measure in a first order theory with NIP and mu(\phi(x,b)) = 0 for all b, then \mu^{(n)}(\exists y(\phi(x_1,y)\wedge ... \wedge \phi(x_n,y))) = 0. We deduce that if G is an fsg grooup then…

We try to understand complete types over a somewhat saturated model of a complete first order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory…

We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global…

We study stable like behaviour in first order theories without the independence property. We introduce generically stable measures, give characterizatiions, and show their ubiquity. We also introduce generic compact domination. We also…

This paper gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates. As a main theorem we show that Presburger arithmetic is closed under unary counting quantifiers.…

I study definable sets in affine continuous logic. Let $T$ be an affine theory. After giving some general results, it is proved that if $T$ has a first order model, its extremal theory is a complete first order theory and first order…

This paper is concerned with the question of when a theory is refutable with certainty on the basis of sequence of primitive observations. Beginning with the simple definition of falsifiability as the ability to be refuted by some finite…

We prove an explicit finite-sample version of the Borel--Cantelli lemma under $m$-dependence. Given any $m$-dependent sequence of events $(A_k)_{1\leq k\leq N}$, we show that \[ \mathbb{P}\Bigl(\bigcup_{k=1}^N A_k\Bigr) \ge 1 -…

The class of abelian $p$-groups are an example of some very interesting phenomena in computable structure theory. We will give an elementary first-order theory $T_p$ whose models are each bi-interpretable with the disjoint union of an…

The Glivenko--Cantelli theorem is a uniform version of the strong law of large numbers. It states that for every IID sequence of random variables, the empirical measure converges to the underlying distribution (in the sense of uniform…

We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), we prove density of compressible types. Using this, we…

We prove an assortment of results on (commutative and unital) NIP rings, especially $\mathbb{F}_p$-algebras. Let $R$ be a NIP ring. Then every prime ideal or radical ideal of $R$ is externally definable, and every localization $S^{-1}R$ is…

A relevant thesis is that for the family of complete first order theories with NIP (i.e. without the independence property) there is a substantial theory, like the family of stable (and the family of simple) first order theories. We examine…

Let F be a separable uniformly bounded family of measurable functions on a standard measurable space, and let N_{[]}(F,\epsilon,\mu) be the smallest number of \epsilon-brackets in L^1(\mu) needed to cover F. The following are equivalent: 1.…

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized…