The tilting equivalence as a bi-interpretation
We propose a model theoretic interpretation of the theorems about the equivalence between mixed characteristic perfectoid spaces and their tilts.
math.LO↗
arXiv subjects
Pierre Simon
Publications and source records attributed to Pierre Simon.
On Descent and germs
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 simpler proof of descent for stably dominated types in $\ACVF$. Furthermore, we demonstrate that any stable set in an $\NIP$ theory has the bounded stabilizing property. This result is subsequently used to correct Proposition 6.7 from the book on stable domination and independence in $\ACVF$.
math.LO↗
Generic Stability Independence and Treeless theories
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 well-behaved. We show that the class of treeless theories contains both binary theories and stable theories and give several applications of the theory of independence for treeless theories. As a corollary, we show that every binary NSOP$_{3}$ theory is simple.
math.LO↗
Residue field domination in some henselian valued fields
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 alternate characterization of stable domination in algebraically closed valued fields for types over parameters in the field sort.
math.LO↗
On large externally definable sets in NIP
We study cofinal systems of finite subsets of $\omega_1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: in an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.
math.LO↗
Model Checking on Interpretations of Classes of Bounded Local Cliquewidth
We present a fixed-parameter tractable algorithm for first-order model checking on interpretations of graph classes with bounded local cliquewidth. Notably, this includes interpretations of planar graphs, and more generally, of classes of bounded genus. To obtain this result we develop a new tool which works in a very general setting of dependent classes and which we believe can be an important ingredient in achieving similar results in the future.
cs.DS↗
Density of compressible types and some consequences
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 obtain explicit uniform honest definitions for NIP formulas (answering a question of Eshel and the second author), and build compressible models in countable NIP theories.
math.LO↗
Dependent finitely homogneneous rosy structures
We study finitely homogeneous dependent rosy structures, adapting results of Cherlin, Harrington, and Lachlan proved for $\omega$-stable $\omega$-categorical structures. In particular, we prove that such structures have finite {\th}-rank and are coordinatized by a {\th}-rank 1 set. We show that they admit a distal, finitely axiomatizable, expansion. These results show that there are, up to inter-definability, at most countably many dependent rosy structures M which are homogeneous in a finite relational language.
math.LO↗
A note on stability and NIP in one variable
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).
math.LO↗
Ordered graphs of bounded twin-width
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, we prove a grid theorem for classes of ordered graphs which have unbounded twin-width. From this we derive that the model-checking problem for first-order logic is fixed-parameter tractable over a hereditary class of ordered graphs if, and -- under common complexity-theoretic assumptions -- only if the class has bounded twin-width. For hereditary classes of ordered graphs, we show that bounded twin-width is equivalent to the NIP property from model theory, as well as the smallness condition from enumerative combinatorics. We prove the existence of a gap in the growth of hereditary classes of ordered graphs. Furthermore, we provide a grid theorem which applies to all monadically NIP classes of structures (ordered or unordered), or equivalently, classes which do not transduce the class of all finite graphs.
cs.LO↗
Twin-width IV: ordered graphs and matrices
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 (hereditary) class of matrices over a finite alphabet either contains at least $n!$ matrices of size $n \times n$, or at most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class over a finite alphabet, answers our small conjecture [SODA '21] in the case of ordered graphs, and with more work, settles a question first asked by Balogh, Bollob\'as, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes of ordered graphs. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width.
math.CO↗
Definable Equivariant Retractions in Non-Archimedean Geometry
For $G$ an algebraic group definable over a model of $\operatorname{ACVF}$, or more generally a definable subgroup of an algebraic group, we study the stable completion $\widehat{G}$ of $G$, as introduced by Loeser and the second author. For $G$ connected and stably dominated, assuming $G$ commutative or that the valued field is of equicharacteristic 0, we construct a pro-definable $G$-equivariant strong deformation retraction of $\widehat{G}$ onto the generic type of $G$. For $G=S$ a semiabelian variety, we construct a pro-definable $S$-equivariant strong deformation retraction of $\widehat{S}$ onto a definable group which is internal to the value group. We show that, in case $S$ is defined over a complete valued field $K$ with value group a subgroup of $\mathbb{R}$, this map descends to an $S(K)$-equivariant strong deformation retraction of the Berkovich analytification $S^{\mathrm{an}}$ of $S$ onto a piecewise linear group, namely onto the skeleton of $S^{\mathrm{an}}$. This yields a construction of such a retraction without resorting to an analytic (non-algebraic) uniformization of $S$. Furthermore, we prove a general result on abelian groups definable in an NIP theory: any such group $G$ is a directed union of $\infty$-definable subgroups which all stabilize a generically stable Keisler measure on $G$.
math.LO↗
Dp and other minimalities
A first order expansion of $(\mathbb{R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, $p$-adic fields, ordered abelian groups with only finitely many convex subgroups (in articular archimedean ordered abelian groups), and abelian groups equipped with archimedean cyclic group orders. The latter allows us to describe unary definable sets in dp-minimal expansions of $(\mathbb{Z},+,C)$, where $C$ is a cyclic group order. Along the way we describe unary definable sets in dp-minimal expansions of ordered abelian groups. In the last section we give a canonical correspondence between dp-minimal expansions of $(\mathbb{Q},+,<)$ and o-minimal expansions $\mathcal{R}$ of $(\mathbb{R},+,<)$ such that $(\mathcal{R},\mathbb{Q})$ is a "dense pair".
math.LO↗
On omega-categorical structures with few finite substructures
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 showed that for primitive actions, the growth is at least exponential (if the sequence is not constant equal to 1). The best lower bound previously known for the base of the exponential was obtained by Merola. We establishing the optimal value of 2 in the case where the structure is unstable. This allows us to improve on Merola's bound and also obtain the optimal value for structures homogeneous in a finite relational language. Finally, we show that the study of sequences (f_n) of sub-exponential growth reduces to the omega-stable case.
math.LO↗
Linear orders in NIP structures
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 interprets an infinite linear order. This partially answers a longstanding open question.
math.LO↗
NIP omega-categorical structures: the rank 1 case
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 many such structures and that all are built out of finitely many linear orders interacting in a restricted number of ways. As an example of application, we deduce the classification of primitive structures homogeneous in a language consisting of n linear orders as well as all reducts of such structures.
math.LO↗
The classification of homogeneous finite-dimensional permutation structures
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 a question of Cameron, and confirming the classification conjectured by the first author. The primitive case was proven by the second author using model-theoretic methods, and those methods continue to appear here.
math.LO↗
Automorphism groups of finite topological rank
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 show how finite topological rank of the automorphism group of an $\omega$-categorical structure can go down to reducts. Together, those results prove that a large number of $\omega$-categorical structures that appear in the literature have an automorphism group of finite topological rank. In fact, we are not aware of any $\omega$-categorical structure to which they do not apply (assuming the automorphism group has no compact quotients). We end with a few questions and conjectures.
math.GR↗