Publications (51)
Distal and non-distal NIP theories
Pierre Simon
We study one way in which stable phenomena can exist in an NIP theory. We start by defining a notion of 'pure instability' that we call 'distality' in which no such phenomenon occu…
Dp-minimal valued fields
Franziska Jahnke, Pierre Simon, Erik Walsberg
We show that dp-minimal valued fields are henselian and that a dp-minimal field admitting a definable type V topology is either real closed, algebraically closed or admits a non-tr…
Groups and fields with NTP2
Artem Chernikov, Itay Kaplan, Pierre Simon
NTP2 is a large class of first-order theories defined by Shelah and generalizing simple and NIP theories. Algebraic examples of NTP2 structures are given by ultra-products of p-adi…
Ordered graphs of bounded twin-width
Pierre Simon, Szymon ToruÅczyk
We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular…
Definably amenable NIP groups
Artem Chernikov, Pierre Simon
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications…
Some NIP-like phenomena in NTP
Itay Kaplan, Pierre Simon
We introduce the notion of an NTP-smooth measure and prove that they exist assuming NTP. Using this, we propose a notion of distality in NTP that unfortunately do…
Dp and other minimalities
Pierre Simon, Erik Walsberg
A first order expansion of is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, -adic fields, ord…
Model Checking on Interpretations of Classes of Bounded Local Cliquewidth
Ãdouard Bonnet, Jan Dreier, Jakub Gajarský +4
We present a fixed-parameter tractable algorithm for first-order model checking on interpretations of graph classes with bounded local cliquewidth. Notably, this includes interpret…
Automorphism groups of finite topological rank
Itay Kaplan, Pierre Simon
We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then…
On omega-categorical structures with few finite substructures
Pierre Simon
We establish new results on the possible growth rates for the sequence (f_n) counting the number of orbits of a given oligomorphic group on unordered sets of size n. Macpherson sho…
On forking and definability of types in some dp-minimal theories
Pierre Simon, Sergei Starchenko
We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst non-forking types.
The tilting equivalence as a bi-interpretation
Silvain Rideau-Kikuchi, Thomas Scanlon, Pierre Simon
We propose a model theoretic interpretation of the theorems about the equivalence between mixed characteristic perfectoid spaces and their tilts.
Definable Equivariant Retractions in Non-Archimedean Geometry
Martin Hils, Ehud Hrushovski, Pierre Simon
For an algebraic group definable over a model of , or more generally a definable subgroup of an algebraic group, we study the stable completion $\widehat{G…
A note on generically stable measures and fsg groups
Ehud Hrushovski, Anand Pillay, Pierre Simon
We prove that if μis a generically stable stable measure in a first order theory with NIP and mu(Ï(x,b)) = 0 for all b, then μ^{(n)}(\exists y(Ï(x_1,y)\wedge ... \wedge Ï(x_n,…
Rosenthal compacta and NIP formulas
Pierre Simon
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 i…
The classification of homogeneous finite-dimensional permutation structures
Samuel Braunfeld, Pierre Simon
We classify the homogeneous finite-dimensional permutation structures, i.e., homogeneous structures in a language of finitely many linear orders, giving a nearly complete answer to…
Externally definable sets and dependent pairs II
Artem Chernikov, Pierre Simon
We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite…
Definable and invariant types in enrichments of NIP theories
Silvain Rideau, Pierre Simon
Let T be an NIP L-theory and T' be an enrichment. We give a sufficient condition on T' for the underlying L-type of any definable (respectively invariant) type over a model of T' t…
Adding linear orders
Saharon Shelah, Pierre Simon
We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Othe…
Stabilizers, groups with f-generics in NTP2 and PRC fields
Samaria Montenegro, Alf Onshuus, Pierre Simon
In this paper we develop three different subjects. We study and prove alternative versions of Hrushovski's "Stabilizer Theorem", we generalize part of the basic theory of definably…
Externally definable sets and dependent pairs
Artem Chernikov, Pierre Simon
We prove that externally definable sets in first order NIP theories have honest definitions, giving a new proof of Shelah's expansion theorem. Also we discuss a weak notion of stab…
Invariant types in NIP theories
Pierre Simon
We study invariant types in NIP theories. Amongst other things: we prove a definable version of the (p,q)-theorem in theories of small or medium directionality; we construct a cano…
Exact saturation in simple and NIP theories
Itay Kaplan, Saharon Shelah, Pierre Simon
A theory is said to have exact saturation at a singular cardinal if it has a -saturated model which is not -saturated. We show, under some set-theoretic assump…
Naming an indiscernible sequence in NIP theories
Artem Chernikov, Pierre Simon
In this short note we show that if we add predicate for a dense complete indiscernible sequence in a dependent theory then the result is still dependent. This answers a question of…
External definability and groups in NIP theories
Artem Chernikov, Anand Pillay, Pierre Simon
We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G^{00}, etc., are preserved when passing to the theory of the She…
A Note on "Regularity lemma for distal structures"
Pierre Simon
In a recent paper, Chernikov and Starchenko prove that graphs defined in distal theories have strong regularity properties, generalizing previous results about graphs defined by se…
The affine and projective groups are maximal
Itay Kaplan, Pierre Simon
We show that the groups AGL_n(Q) and PGL_n(Q), seen as closed subgroups of S_{\infty}, are maximal-closed.
Witnessing dp-rank
Itay Kaplan, Pierre Simon
We prove that in NTP_2 theories if p is a dependent type with dp-rank >= κ, then this can be witnessed by indiscernible sequences of tuples satisfying p. If p has dp-rank infinity…
On Descent and germs
Pierre Simon, Mariana Vicaria
We present a new proof of descent for stably dominated types in any theory, dropping the hypothesis of the existence of global invariant extensions. Additionally, we give a much si…
Dependent finitely homogneneous rosy structures
Alf Onshuus, Pierre Simon
We study finitely homogeneous dependent rosy structures, adapting results of Cherlin, Harrington, and Lachlan proved for -stable -categorical structures. In particular, we…
Twin-width IV: ordered graphs and matrices
Ãdouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez +3
We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a…
On large externally definable sets in NIP
Martin Bays, Omer Ben-Neria, Itay Kaplan +1
We study cofinal systems of finite subsets of . We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a quest…
Generic Stability Independence and Treeless theories
Itay Kaplan, Nicholas Ramsey, Pierre Simon
We initiate a systematic study of \emph{generic stability independence} and introduce the class of \emph{treeless theories} in which this notion of independence is particularly wel…
Type decomposition in NIP theories
Pierre Simon
We prove that any type in an NIP theory can be decomposed into a stable part (a generically stable partial type) and a distal-like quotient.
Finding generically stable measures
Pierre Simon
We discuss two constructions for obtaining generically stable Keisler measures in an NIP theory. First, we show how to symmetrize an arbitrary invariant measure to obtain a generic…
Dp-minimality: invariant types and dp-rank
Pierre Simon
This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p…
Generically stable and smooth measures in NIP theories
Ehud Hrushovski, Anand Pillay, Pierre Simon
We study stable like behaviour in first order theories without the independence property. We introduce generically stable measures, give characterizatiions, and show their ubiquity…
Linear orders in NIP structures
Pierre Simon
We show that every unstable NIP theory admits a V-definable linear quasi-order, over a finite set of parameters. In particular, if the theory is omega-categorical, then it interpre…
Some remarks on dp-minimal groups
Elad Levi, Itay Kaplan, Pierre Simon
We prove that -categorical dp-minimal groups are nilpotent-by-finite. We also show that in dp-minimal definably amenable groups, f-generic global types are strongly f-gen…
Tame Topology over Dp-minimal Structures
Pierre Simon, Erik Walsberg
We develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets ar…
NIP henselian valued fields
Franziska Jahnke, Pierre Simon
We show that any theory of tame henselian valued fields is NIP if and only if the theory of its residue field and the theory of its value group are NIP. Moreover, we show that if $…
NIP omega-categorical structures: the rank 1 case
Pierre Simon
We classify primitive, rank 1, omega-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely man…
A note on stability and NIP in one variable
Pierre Simon
A theory is NIP (resp. stable) if and only if every formula with parameters in two single variables is NIP (resp. does not have the order property).
Boundedness and absoluteness of some dynamical invariants in model theory
Krzysztof Krupinski, Ludomir Newelski, Pierre Simon
Let be a monster model of an arbitrary theory , any tuple of bounded length of elements of , and an enumeration of all elements…
The Borel cardinality of Lascar strong types
Itay Kaplan, Benjamin Miller, Pierre Simon
We show that if the restriction of the Lascar equivalence relation to a KP-strong type is non-trivial, then it is non-smooth (when viewed as a Borel equivalence relation on an appr…
Henselian valued fields and inp-minimality
Artem Chernikov, Pierre Simon
We prove that every ultraproduct of -adics is inp-minimal (i.e., of burden ). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselia…
VC-sets and generic compact domination
Pierre Simon
Let X be a closed subset of a locally compact second countable group G whose family of translates has finite VC-dimension. We show that the topological border of X has Haar measure…
A Guide to NIP theories
Pierre Simon
This text is an introduction to the study of NIP (or dependent) theories. It is meant to serve two purposes. The first is to present various aspects of NIP theories and give the re…
Density of compressible types and some consequences
Martin Bays, Itay Kaplan, Pierre Simon
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…
On amalgamation in NTP2 theories and generically simple generics
Pierre Simon
We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This impr…
Residue field domination in some henselian valued fields
Clifton Ealy, Deirdre Haskell, Pierre Simon
We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an…