Abstract In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called āuniform definability of types over finite setsā ā¦
Abstract In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called āuniform definability of types over finite setsā (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density ā¦
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density and dp-rank coincide in the natural way.
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density ā¦
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density and dp-rank coincide in the natural way.
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they ā¦
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they are not unique but admit a quasi-ordering under which the minimal decomposition is unique. Both cases lead to a one-dimensional elimination of imaginaries in VC-minimal and quasi-VC-minimal theories.
In this paper, we study VapnikāChervonenkis density (VCādensity) over indiscernible sequences (denoted VC ind ādensity). We answer an open question in [1], showing that VC ind ādensity is always integer ā¦
In this paper, we study VapnikāChervonenkis density (VCādensity) over indiscernible sequences (denoted VC ind ādensity). We answer an open question in [1], showing that VC ind ādensity is always integer valued. We also show that VC ind ādensity and dpārank coincide in the natural way.
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex ā¦
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.
Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that ā¦
Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p -adics are not quasi-VC-minimal.
In this paper, we show that \phi is a dependent formula if and only if all \phi-types have an extension to a \phi-isolated \phi-type that is an elementary \phi-extension (see ā¦
In this paper, we show that \phi is a dependent formula if and only if all \phi-types have an extension to a \phi-isolated \phi-type that is an elementary \phi-extension (see Definition 2.3 in the paper). Moreover, we show that the domain of this extension adds at most 2 times the independence dimension of \phi new elements to the domain of the original \phi-type. We give corollaries to this theorem and discuss parallels to the stable setting.
In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called uniform definability of types over finite sets (UDTFS). ā¦
In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called uniform definability of types over finite sets (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.
We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse ā¦
We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse of function-space indiscernibles (i.e. parameterized equivalence relations) to characterize rosy theories; and finally, convex equivalence relation indiscernibles to characterize NTP2 theories.
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they ā¦
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they are not unique but admit a quasi-ordering under which the minimal decomposition is unique. Both cases lead to a one-dimensional elimination of imaginaries in VC-minimal and quasi-VC-minimal theories.
In this paper, we show that a partitioned formula Ļis dependent if and only if Ļhas uniform definability of types over finite partial order indiscernibles. This generalizes our result from ā¦
In this paper, we show that a partitioned formula Ļis dependent if and only if Ļhas uniform definability of types over finite partial order indiscernibles. This generalizes our result from a previous paper [1]. We show this by giving a decomposition of the truth values of an externally definable formula on a finite partial order indiscernible.
We show VC-minimality is $\Pi ^0_4$-complete. In particular, we give a local characterization of VC-minimality. We also show dp-smallness is $\Pi ^1_1$-complete.
We show VC-minimality is $\Pi ^0_4$-complete. In particular, we give a local characterization of VC-minimality. We also show dp-smallness is $\Pi ^1_1$-complete.
We show that any formula with two free variables in a VapnikāChervonenkis (VC) minimal theory has VC-codensity at most 2. Modifying the argument slightly, we give a new proof of ā¦
We show that any formula with two free variables in a VapnikāChervonenkis (VC) minimal theory has VC-codensity at most 2. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acleq= dcleq, the VC-codensity of a formula is at most the number of free variables (from the work of Aschenbrenner et al., the author, and Laskowski).
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing ā¦
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing the order property, and use this to create op-rank, which generalizes 2-rank. From this we build op-dimension. We show that op-dimension bounds dp-rank, that op-dimension is sub-additive, and op-dimension generalizes o-minimal dimension in o-minimal theories.
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this ā¦
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, ā¦
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.
We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraisse classes and by complete prime filter classes. ā¦
We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraisse classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.
We prove that, for any $d$ linearly independent functions from some set into a $d$-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination ā¦
We prove that, for any $d$ linearly independent functions from some set into a $d$-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination of these functions has VC-dimension and Littlestone dimension $d-1$. Additionally, we characterize when such families are maximal of VC-dimension $d-1$ and give a sufficient condition for when they are maximal of Littlestone dimension $d-1$.
We prove that, for any $d$ linearly independent functions from some set into a $d$-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination ā¦
We prove that, for any $d$ linearly independent functions from some set into a $d$-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination of these functions has VC-dimension and Littlestone dimension $d-1$. Additionally, we characterize when such families are maximal of VC-dimension $d-1$ and give a sufficient condition for when they are maximal of Littlestone dimension $d-1$.
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density ā¦
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density and dp-rank coincide in the natural way.
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this ā¦
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they ā¦
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they are not unique but admit a quasi-ordering under which the minimal decomposition is unique. Both cases lead to a one-dimensional elimination of imaginaries in VC-minimal and quasi-VC-minimal theories.
In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). ā¦
In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.
We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse ā¦
We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse of function-space indiscernibles (i.e. parameterized equivalence relations) to characterize rosy theories; and finally, convex equivalence relation indiscernibles to characterize NTP2 theories.
In this paper, we show that \phi is a dependent formula if and only if all \phi-types have an extension to a \phi-isolated \phi-type that is an "elementary \phi-extension" (see ā¦
In this paper, we show that \phi is a dependent formula if and only if all \phi-types have an extension to a \phi-isolated \phi-type that is an "elementary \phi-extension" (see Definition 2.3 in the paper). Moreover, we show that the domain of this extension adds at most 2 times the independence dimension of \phi new elements to the domain of the original \phi-type. We give corollaries to this theorem and discuss parallels to the stable setting.
We show that any formula with two free variables in a VC-minimal theory has VC-codensity at most two. Modifying the argument slightly, we give a new proof of the fact ā¦
We show that any formula with two free variables in a VC-minimal theory has VC-codensity at most two. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acl = dcl, the VC-codensity of a formula is at most the number of free variables.
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex ā¦
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.
We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraisse classes and by complete prime filter classes. ā¦
We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraisse classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, ā¦
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing ā¦
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing the order property, and use this to create op-rank, which generalizes 2-rank. From this we build op-dimension. We show that op-dimension bounds dp-rank, that op-dimension is sub-additive, and op-dimension generalizes o-minimal dimension in o-minimal theories.
We show that any formula with two free variables in a VapnikāChervonenkis (VC) minimal theory has VC-codensity at most 2. Modifying the argument slightly, we give a new proof of ā¦
We show that any formula with two free variables in a VapnikāChervonenkis (VC) minimal theory has VC-codensity at most 2. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acleq= dcleq, the VC-codensity of a formula is at most the number of free variables (from the work of Aschenbrenner et al., the author, and Laskowski).
We prove that, for any $d$ linearly independent functions from some set into a $d$-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination ā¦
We prove that, for any $d$ linearly independent functions from some set into a $d$-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination of these functions has VC-dimension and Littlestone dimension $d-1$. Additionally, we characterize when such families are maximal of VC-dimension $d-1$ and give a sufficient condition for when they are maximal of Littlestone dimension $d-1$.
We prove that, for any $d$ linearly independent functions from some set into a $d$-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination ā¦
We prove that, for any $d$ linearly independent functions from some set into a $d$-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination of these functions has VC-dimension and Littlestone dimension $d-1$. Additionally, we characterize when such families are maximal of VC-dimension $d-1$ and give a sufficient condition for when they are maximal of Littlestone dimension $d-1$.
We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraisse classes and by complete prime filter classes. ā¦
We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraisse classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.
We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraisse classes and by complete prime filter classes. ā¦
We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraisse classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.
We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse ā¦
We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse of function-space indiscernibles (i.e. parameterized equivalence relations) to characterize rosy theories; and finally, convex equivalence relation indiscernibles to characterize NTP2 theories.
We show VC-minimality is $\Pi ^0_4$-complete. In particular, we give a local characterization of VC-minimality. We also show dp-smallness is $\Pi ^1_1$-complete.
We show VC-minimality is $\Pi ^0_4$-complete. In particular, we give a local characterization of VC-minimality. We also show dp-smallness is $\Pi ^1_1$-complete.
We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse ā¦
We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse of function-space indiscernibles (i.e. parameterized equivalence relations) to characterize rosy theories; and finally, convex equivalence relation indiscernibles to characterize NTP2 theories.
Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that ā¦
Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p -adics are not quasi-VC-minimal.
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they ā¦
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they are not unique but admit a quasi-ordering under which the minimal decomposition is unique. Both cases lead to a one-dimensional elimination of imaginaries in VC-minimal and quasi-VC-minimal theories.
In this paper, we study VapnikāChervonenkis density (VCādensity) over indiscernible sequences (denoted VC ind ādensity). We answer an open question in [1], showing that VC ind ādensity is always integer ā¦
In this paper, we study VapnikāChervonenkis density (VCādensity) over indiscernible sequences (denoted VC ind ādensity). We answer an open question in [1], showing that VC ind ādensity is always integer valued. We also show that VC ind ādensity and dpārank coincide in the natural way.
We show that any formula with two free variables in a VC-minimal theory has VC-codensity at most two. Modifying the argument slightly, we give a new proof of the fact ā¦
We show that any formula with two free variables in a VC-minimal theory has VC-codensity at most two. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acl = dcl, the VC-codensity of a formula is at most the number of free variables.
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this ā¦
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing ā¦
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing the order property, and use this to create op-rank, which generalizes 2-rank. From this we build op-dimension. We show that op-dimension bounds dp-rank, that op-dimension is sub-additive, and op-dimension generalizes o-minimal dimension in o-minimal theories.
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this ā¦
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing ā¦
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing the order property, and use this to create op-rank, which generalizes 2-rank. From this we build op-dimension. We show that op-dimension bounds dp-rank, that op-dimension is sub-additive, and op-dimension generalizes o-minimal dimension in o-minimal theories.
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex ā¦
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.
Abstract In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called āuniform definability of types over finite setsā ā¦
Abstract In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called āuniform definability of types over finite setsā (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex ā¦
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they ā¦
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they are not unique but admit a quasi-ordering under which the minimal decomposition is unique. Both cases lead to a one-dimensional elimination of imaginaries in VC-minimal and quasi-VC-minimal theories.
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, ā¦
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density ā¦
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density and dp-rank coincide in the natural way.
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density ā¦
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density and dp-rank coincide in the natural way.
In this paper, we show that a partitioned formula Ļis dependent if and only if Ļhas uniform definability of types over finite partial order indiscernibles. This generalizes our result from ā¦
In this paper, we show that a partitioned formula Ļis dependent if and only if Ļhas uniform definability of types over finite partial order indiscernibles. This generalizes our result from a previous paper [1]. We show this by giving a decomposition of the truth values of an externally definable formula on a finite partial order indiscernible.
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density ā¦
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density and dp-rank coincide in the natural way.
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they ā¦
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they are not unique but admit a quasi-ordering under which the minimal decomposition is unique. Both cases lead to a one-dimensional elimination of imaginaries in VC-minimal and quasi-VC-minimal theories.
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, ā¦
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.
In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called uniform definability of types over finite sets (UDTFS). ā¦
In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called uniform definability of types over finite sets (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.
In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). ā¦
In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.
In this paper, we show that \phi is a dependent formula if and only if all \phi-types have an extension to a \phi-isolated \phi-type that is an elementary \phi-extension (see ā¦
In this paper, we show that \phi is a dependent formula if and only if all \phi-types have an extension to a \phi-isolated \phi-type that is an elementary \phi-extension (see Definition 2.3 in the paper). Moreover, we show that the domain of this extension adds at most 2 times the independence dimension of \phi new elements to the domain of the original \phi-type. We give corollaries to this theorem and discuss parallels to the stable setting.
In this paper, we show that \phi is a dependent formula if and only if all \phi-types have an extension to a \phi-isolated \phi-type that is an "elementary \phi-extension" (see ā¦
In this paper, we show that \phi is a dependent formula if and only if all \phi-types have an extension to a \phi-isolated \phi-type that is an "elementary \phi-extension" (see Definition 2.3 in the paper). Moreover, we show that the domain of this extension adds at most 2 times the independence dimension of \phi new elements to the domain of the original \phi-type. We give corollaries to this theorem and discuss parallels to the stable setting.
We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of ā¦
We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.
Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and ā¦
Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and any theory of pure tree is dp-minimal.
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, ā¦
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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal theories.
Abstract We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable ā¦
Abstract We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M , generalizing a result of Dolich on o-minimal theories in [4].
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 ā¦
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 is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first, we should try to understand the quite saturated case). As a measure of our progress, we give several applications considering some test questions; in particular, we try to prove the generic pair conjecture and do it for measurable cardinals.
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its ā¦
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types, and we discuss the possible relations between dp-rank and VC-density.
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they ā¦
Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they are not unique but admit a quasi-ordering under which the minimal decomposition is unique. Both cases lead to a one-dimensional elimination of imaginaries in VC-minimal and quasi-VC-minimal theories.
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized as studied in special cases in (for example) [7], [9]. We understand of ā¦
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized as studied in special cases in (for example) [7], [9]. We understand of indiscernibles to be special kinds of companionable theories of finite structures, and much of the work in our arguments is carried in the context of the model-companion. Among other things, this approach allows us to prove that the companion of a theory of whose base consists of the quantifier-free formulas is necessarily the theory of the Fraisse limit of a Fraisse class of linearly ordered finite structures (where the linear order will be at least quantifier-free definable). We also provide streamlined arguments for the result of [6] identifying extremely amenable groups with the automorphism groups of limits of Ramsey classes.
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be strong amalgamation classes of finite structures, with disjoint finite signatures $\sigma$ and $\tau$. Then $\mathcal{C}_1 \wedge \mathcal{C}_2$ denotes the class of all finite ($\sigma\cup\tau$)-structures whose ā¦
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be strong amalgamation classes of finite structures, with disjoint finite signatures $\sigma$ and $\tau$. Then $\mathcal{C}_1 \wedge \mathcal{C}_2$ denotes the class of all finite ($\sigma\cup\tau$)-structures whose $\sigma$-reduct is from $\mathcal{C}_1$ and whose $\tau$-reduct is from $\mathcal{C}_2$. We prove that when $\mathcal{C}_1$ and $\mathcal{C}_2$ are Ramsey, then $\mathcal{C}_1 \wedge \mathcal{C}_2$ is also Ramsey. We also discuss variations of this statement, and give several examples of new Ramsey classes derived from those general results.
Abstract Dp-minimality is a common generalization of weak minimality and weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal (Fact 2.2), but there are dp-minimal densely ā¦
Abstract Dp-minimality is a common generalization of weak minimality and weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal (Fact 2.2), but there are dp-minimal densely ordered groups that are not weakly o-minimal. We introduce the even more general notion of inp-minimality and prove that in an inp-minimal densely ordered group, every definable unary function is a union of finitely many continuous locally monotonic functions (Theorem 3.2).
We study the VapnikāChervonenkis (VC) density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite U-rank ā¦
We study the VapnikāChervonenkis (VC) density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite U-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.
Abstract We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight.
Abstract We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight.
A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of ā¦
A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be closely related to modular pairs in the lattice of algebraically closed sets.
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density ā¦
In this paper, we study VC-density over indiscernible sequences (denoted VC_ind-density). We answer an open question in [1], showing that VC_ind-density is always integer valued. We also show that VC_ind-density and dp-rank coincide in the natural way.
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of ā¦
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of its dp-minimal types, and discuss the possible relations between dp-rank and VC-density.
Let C_1 and C_2 be strong amalgamation classes of finite structures, with disjoint finite signatures sigma and tau. Then C_1 wedge C_2 denotes the class of all finite (sigma cup ā¦
Let C_1 and C_2 be strong amalgamation classes of finite structures, with disjoint finite signatures sigma and tau. Then C_1 wedge C_2 denotes the class of all finite (sigma cup tau)-structures whose sigma-reduct is from C_1 and whose tau-reduct is from C_2. We prove that when C_1 and C_2 are Ramsey, then C_1 wedge C_2 is also Ramsey. We also discuss variations of this statement, and give several examples of new Ramsey classes derived from those general results.
We show that a class of subsets of a structure uniformly definable by a first-order formula is a Vapnik-Chervonenkis class if and only if the formula does not have the ā¦
We show that a class of subsets of a structure uniformly definable by a first-order formula is a Vapnik-Chervonenkis class if and only if the formula does not have the independence property. Via this connection we obtain several new examples of Vapnik-Chervonenkis classes, including sets of positivity of finitely subanalytic functions.
Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that ā¦
Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p -adics are not quasi-VC-minimal.
It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain ā¦
It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in Kn of certain definable R-submodules of Kn (for all ). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts.
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex ā¦
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.
The study of permutation groups has always been closely associated with that of highly symmetric structures. The objects considered here are countably infinite, but have only finitely many different substructures ā¦
The study of permutation groups has always been closely associated with that of highly symmetric structures. The objects considered here are countably infinite, but have only finitely many different substructures of any given finite size. They are precisely those structures which are determined by first-order logical axioms together with the assumption of countability. This book concerns such structures, their substructures and their automorphism groups. A wide range of techniques are used: group theory, combinatorics, Baire category and measure among them. The book arose from lectures given at a research symposium and retains their informal style, whilst including as well many recent results from a variety of sources. It concludes with exercises and unsolved research problems.
Abstract In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called āuniform definability of types over finite setsā ā¦
Abstract In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called āuniform definability of types over finite setsā (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.
In this paper, we study VapnikāChervonenkis density (VCādensity) over indiscernible sequences (denoted VC ind ādensity). We answer an open question in [1], showing that VC ind ādensity is always integer ā¦
In this paper, we study VapnikāChervonenkis density (VCādensity) over indiscernible sequences (denoted VC ind ādensity). We answer an open question in [1], showing that VC ind ādensity is always integer valued. We also show that VC ind ādensity and dpārank coincide in the natural way.
Abstract We examine several conditions, either the existence of a rank or a particular property of Ć¾-forking that suggest the existence of a well-behaved independence relation, and determine the consequences ā¦
Abstract We examine several conditions, either the existence of a rank or a particular property of Ć¾-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.
We present a programme of characterizing Ramsey classes of structures by a combination of the model theory and combinatorics. In particular, we relate the classification programme of countable homogeneous structures ā¦
We present a programme of characterizing Ramsey classes of structures by a combination of the model theory and combinatorics. In particular, we relate the classification programme of countable homogeneous structures (of Lachlan and Cherlin) to the classification of Ramsey classes. As particular instances of this approach we characterize all Ramsey classes of graphs, tournaments and partial ordered sets. We fully characterize all monotone Ramsey classes of relational systems (of any type). We also carefully discuss the role of (admissible) orderings which lead to a new classification of Ramsey properties by means of classes of order-invariant objects.
Abstract Kueker's conjecture is proved for stable theories, for theories that interpret a linear ordering, and for theories with Skolem functions. The proof of the stable case involves certain results ā¦
Abstract Kueker's conjecture is proved for stable theories, for theories that interpret a linear ordering, and for theories with Skolem functions. The proof of the stable case involves certain results on coordinatization that are of independent interest.
Our thesis is that for the family of classes of the form EC(T),T a com- plete first order theory with the dependence property (which is just the negation of the ā¦
Our thesis is that for the family of classes of the form EC(T),T a com- plete first order theory with the dependence property (which is just the negation of the independence property) there is a substantial theory which means: a substantial body of basic results for all such classes and some complimentary results for the first order theories with the independence property, as for the family of stable (and the family of simple) first order theories. We examine some properties.
Abstract We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable ā¦
Abstract We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming āSwiss cheesesā in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in āmostā valued fields F , if f ( x ), g ( x ) ā F [ x ] and v is the valuation map, then the set { x : v ( f ( x )) ā¤ v ( g ( x ))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of āvalued treesā, which we define formally.
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It was shown in \cite{sc12} that for a certain class of structures $\I$, $\I$-indexed indiscernible sets have the modeling property just in case the age of $\I$ is a Ramsey ā¦
It was shown in \cite{sc12} that for a certain class of structures $\I$, $\I$-indexed indiscernible sets have the modeling property just in case the age of $\I$ is a Ramsey class. We expand this known class of structures from ordered structures in a finite relational language to ordered, locally finite structures which isolate quantifier-free types by way of quantifier-free formulas. As a corollary, we may conclude that certain classes of finite trees are Ramsey, some previously known. See updated paper for new references.
We prove a tight bound on the number of realized 0=1 patterns (or equivalently on the VapnikāChervonenkis codensity) of definable families in models of the theory of algebraically closed valued ā¦
We prove a tight bound on the number of realized 0=1 patterns (or equivalently on the VapnikāChervonenkis codensity) of definable families in models of the theory of algebraically closed valued fields with a non-archimedean valuation. Our result improves the best known result in this direction proved by Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko, who proved a weaker bound in the restricted case where the characteristics of the field K and its residue field are both assumed to be 0. The bound obtained here is optimal and without any restriction on the characteristics. We obtain the aforementioned bound as a consequence of another result on bounding the Betti numbers of semi-algebraic subsets of certain Berkovich analytic spaces, mirroring similar results known already in the case of o-minimal structure and for real closed as well as algebraically closed fields. The latter result is the first result in this direction and is possibly of independent interest. Its proof relies heavily on recent results of Hrushovski and Loeser on the topology of semi-algebraic subsets of Berkovich analytic spaces.
We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler's order. Thus Keisler's order is infinite and not a well order. Moreover, ā¦
We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler's order. Thus Keisler's order is infinite and not a well order. Moreover, this chain occurs within the simple unstable theories, considered model-theoretically tame. Keisler's order is a central notion of the model theory of the 60s and 70s which compares first-order theories (and implicitly ultrafilters) according to saturation of ultrapowers. Prior to this paper, it was long thought to have finitely many classes, linearly ordered. The model-theoretic complexity we find is witnessed by a very natural class of theories, the $n$-free $k$-hypergraphs studied by Hrushovski. This complexity reflects the difficulty of amalgamation and appears orthogonal to forking.
An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is ā¦
An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is characterised in terms of modular pairs in the lattice of algebraically closed sets. Wherever possible, forking and thorn-forking are treated in a uniform way. They are dual in the sense that forking is the finest (most restrictive) and thorn-forking the coarsest independence relation worth examining. We finish by defining the kernel of a sequence of indiscernibles and studying its relation to canonical bases.
A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures ā¦
A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.