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 …
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 …
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 Vapnik‐Chervonenkis density (VC‐density) over indiscernible sequences (denoted VC ind ‐density). We answer an open question in [1], showing that VC ind ‐density is always integer …
In this paper, we study Vapnik‐Chervonenkis density (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.
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 …
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.
We examine one consequence for the generic theory $T_\mathbf {C}$ of a geometric Fraïssé class $\mathbf {C}$ when $\mathbf {C}$ has the $0,1$-law for first-order logic with convergence to $T_\mathbf …
We examine one consequence for the generic theory $T_\mathbf {C}$ of a geometric Fraïssé class $\mathbf {C}$ when $\mathbf {C}$ has the $0,1$-law for first-order logic with convergence to $T_\mathbf {C}$ itself. We show that in this scenario, if the asymp
We analyze the notion of weak elimination of hyperimaginaries (WEHI) in simple theories. A key observation in the analysis is a characterization of WEHI in terms of forking dependence -- …
We analyze the notion of weak elimination of hyperimaginaries (WEHI) in simple theories. A key observation in the analysis is a characterization of WEHI in terms of forking dependence -- a condition we dub dependence-witnessed-by-imaginaries (DWIP). Generalizing results of [1] and [3], we show that in a simple theory with WEHI, forking and thorn-forking coincide. We also show that, conversely, the equivalence of independence and thorn-independence is (almost) sufficient for WEHI. Thus, the WEHI and the statement independence = thorn-independence are morally equivalent. As a further application of our technology, we demonstrate stable forking for 1-based theories of finite SU-rank that have WEHI.
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 …
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 demonstrate that for the $k$-variable theory $T$ of a finite structure (satisfying certain amalgamation conditions), if finite models of $T$ can be recovered from diagrams of finite {\em subsets} …
We demonstrate that for the $k$-variable theory $T$ of a finite structure (satisfying certain amalgamation conditions), if finite models of $T$ can be recovered from diagrams of finite {\em subsets} of model of $T$ in a certain way, then $T$ is rosy -- in fact, a certain natural $\aleph_0$-categorical completion $T^{\lim}$ of $T$ is super-rosy of finite $U^\thorn$-rank. In an appendix, we also show that any $k$-variable theory $T$ of a finite structure for which the Strong $L^k$-Canonization Problem is efficient soluble has the necessary amalgamation properties up to taking an appropriate reduct.
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 …
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 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 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 give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [7], [9]. We understand …
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [7], [9]. We understand "theories 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 indiscernibles 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.
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 …
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.
We demonstrate that for the $k$-variable theory $T$ of a finite structure (satisfying certain amalgamation conditions), if finite models of $T$ can be recovered from diagrams of finite {\em subsets} …
We demonstrate that for the $k$-variable theory $T$ of a finite structure (satisfying certain amalgamation conditions), if finite models of $T$ can be recovered from diagrams of finite {\em subsets} of model of $T$ in a certain "efficient" way, then $T$ is rosy -- in fact, a certain natural $\aleph_0$-categorical completion $T^{\lim}$ of $T$ is super-rosy of finite $U^\thorn$-rank. In an appendix, we also show that any $k$-variable theory $T$ of a finite structure for which the Strong $L^k$-Canonization Problem is efficient soluble has the necessary amalgamation properties up to taking an appropriate reduct.
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 …
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 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 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 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 …
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 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 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 examine one consequence for the generic theory $T_\mathbf {C}$ of a geometric Fraïssé class $\mathbf {C}$ when $\mathbf {C}$ has the $0,1$-law for first-order logic with convergence to $T_\mathbf …
We examine one consequence for the generic theory $T_\mathbf {C}$ of a geometric Fraïssé class $\mathbf {C}$ when $\mathbf {C}$ has the $0,1$-law for first-order logic with convergence to $T_\mathbf {C}$ itself. We show that in this scenario, if the asymp
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 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 …
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 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 …
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 study Vapnik‐Chervonenkis density (VC‐density) over indiscernible sequences (denoted VC ind ‐density). We answer an open question in [1], showing that VC ind ‐density is always integer …
In this paper, we study Vapnik‐Chervonenkis density (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 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 …
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 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 …
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 demonstrate that for the $k$-variable theory $T$ of a finite structure (satisfying certain amalgamation conditions), if finite models of $T$ can be recovered from diagrams of finite {\em subsets} …
We demonstrate that for the $k$-variable theory $T$ of a finite structure (satisfying certain amalgamation conditions), if finite models of $T$ can be recovered from diagrams of finite {\em subsets} of model of $T$ in a certain way, then $T$ is rosy -- in fact, a certain natural $\aleph_0$-categorical completion $T^{\lim}$ of $T$ is super-rosy of finite $U^\thorn$-rank. In an appendix, we also show that any $k$-variable theory $T$ of a finite structure for which the Strong $L^k$-Canonization Problem is efficient soluble has the necessary amalgamation properties up to taking an appropriate reduct.
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 …
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.
We analyze the notion of weak elimination of hyperimaginaries (WEHI) in simple theories. A key observation in the analysis is a characterization of WEHI in terms of forking dependence -- …
We analyze the notion of weak elimination of hyperimaginaries (WEHI) in simple theories. A key observation in the analysis is a characterization of WEHI in terms of forking dependence -- a condition we dub dependence-witnessed-by-imaginaries (DWIP). Generalizing results of [1] and [3], we show that in a simple theory with WEHI, forking and thorn-forking coincide. We also show that, conversely, the equivalence of independence and thorn-independence is (almost) sufficient for WEHI. Thus, the WEHI and the statement independence = thorn-independence are morally equivalent. As a further application of our technology, we demonstrate stable forking for 1-based theories of finite SU-rank that have WEHI.
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [7], [9]. We understand …
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [7], [9]. We understand "theories 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 indiscernibles 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.
We demonstrate that for the $k$-variable theory $T$ of a finite structure (satisfying certain amalgamation conditions), if finite models of $T$ can be recovered from diagrams of finite {\em subsets} …
We demonstrate that for the $k$-variable theory $T$ of a finite structure (satisfying certain amalgamation conditions), if finite models of $T$ can be recovered from diagrams of finite {\em subsets} of model of $T$ in a certain "efficient" way, then $T$ is rosy -- in fact, a certain natural $\aleph_0$-categorical completion $T^{\lim}$ of $T$ is super-rosy of finite $U^\thorn$-rank. In an appendix, we also show that any $k$-variable theory $T$ of a finite structure for which the Strong $L^k$-Canonization Problem is efficient soluble has the necessary amalgamation properties up to taking an appropriate reduct.
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 …
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 …
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 …
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.
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 …
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.
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 …
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.
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 …
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.
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 …
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 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 …
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.
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 …
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.
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences …
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-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.
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 …
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.
We develop a framework in which Szemerédi's celebrated Regularity Lemma for graphs interacts with core model-theoretic ideas and techniques. Our work relies on a coincidence of ideas from model theory …
We develop a framework in which Szemerédi's celebrated Regularity Lemma for graphs interacts with core model-theoretic ideas and techniques. Our work relies on a coincidence of ideas from model theory and graph theory: arbitrarily large half-graphs coincide with model-theoretic instability, so in their absence, structure theorems and technology from stability theory apply. In one direction, we address a problem from the classical Szemerédi theory. It was known that the "irregular pairs" in the statement of Szemerédi's Regularity Lemma cannot be eliminated, due to the counterexample of half-graphs (i.e., the order property, corresponding to model-theoretic instability). We show that half-graphs are the only essential difficulty, by giving a much stronger version of Szemerédi's Regularity Lemma for models of stable theories of graphs (i.e. graphs with the non-$k_*$-order property), in which there are no irregular pairs, the bounds are significantly improved, and each component satisfies an indivisibility condition. In the other direction, we take a more model-theoretic approach, and give several new Szemerédi-type partition theorems for models of stable theories of graphs. The first theorem gives a partition of any such graph into indiscernible components, meaning here that each component is either a complete or an empty graph, whose interaction is strongly uniform. This relies on a finitary version of the classic model-theoretic fact that stable theories admit large sets of indiscernibles, by showing that in models of stable theories of graphs one can extract much larger indiscernible sets than expected by Ramsey's theorem. The second and third theorems allow for a much smaller number of components at the cost of weakening the "indivisibility" condition on the components. We also discuss some extensions to graphs without the independence property. All graphs are finite and all partitions are equitable, i.e. the sizes of the components differ by at most 1. In the last three theorems, the number of components depends on the size of the graph; in the first theorem quoted, this number is a function of $\epsilon$ only as in the usual Szemerédi Regularity Lemma.
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 …
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.
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, …
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.
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 …
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.
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 …
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.
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 …
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.
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 …
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.
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 …
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.
Abstract We study the structure of Σ 1 1 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h‐reducibility and FF‐reducibility, respectively. We …
Abstract We study the structure of Σ 1 1 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h‐reducibility and FF‐reducibility, respectively. We show that the structure is rich even when one fixes the number of properly \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^1_1\ \big ($\end{document} i.e., Σ 1 1 but not \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Delta ^1_1\big )$\end{document} equivalence classes. We also show the existence of incomparable Σ 1 1 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ 1 1 equivalence relations (under both reducibilities) and show that existence of infinitely many properly Σ 1 1 equivalence classes that are complete as Σ 1 1 sets (under the corresponding reducibility on sets) is necessary but not sufficient for a relation to be complete in the context of Σ 1 1 equivalence relations.
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of …
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure. We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.
Abstract We investigate the geometry of forking for SU-rank 2 elements in supersimple ω-categorical theories and prove stable forking and some structural properties for such elements. We extend this analysis …
Abstract We investigate the geometry of forking for SU-rank 2 elements in supersimple ω-categorical theories and prove stable forking and some structural properties for such elements. We extend this analysis to the case of SU-rank 3 elements.
Abstract For a first-order formula φ(x; y) we introduce and study the characteristic sequence ( P n : n < ω ) of hypergraphs defined by . We show that …
Abstract For a first-order formula φ(x; y) we introduce and study the characteristic sequence ( P n : n < ω ) of hypergraphs defined by . We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theoretic properties of φ and vice versa. The main results are a characterization of NIP and of simplicity in terms of persistence of configurations in the characteristic sequence. Specifically, we show that some tree properties are detected by the presence of certain combinatorial configurations in the characteristic sequence while other properties such as instability and the independence property manifest themselves in the persistence of complicated configurations under localization.
Abstract The following question is open: Does there exist a hyperarithmetic class of computable structures with exactly one non-hyperarithmetic isomorphism-type? Given any oracle α ∈ 2 ω , we can …
Abstract The following question is open: Does there exist a hyperarithmetic class of computable structures with exactly one non-hyperarithmetic isomorphism-type? Given any oracle α ∈ 2 ω , we can ask the same question relativized to α . A negative answer for every α implies Vaught's Conjecture for L ω 1 ω .
Abstract In [3]. two different effective versions of Borel embedding are defined. The first, called computable embedding , is based on uniform enumeration reducibility. while the second, called Turing computable …
Abstract In [3]. two different effective versions of Borel embedding are defined. The first, called computable embedding , is based on uniform enumeration reducibility. while the second, called Turing computable embedding , is based on uniform Turing reducibility. While [3] focused mainly on computable embeddings, the present paper considers Turing computable embeddings. Although the two notions are not equivalent, we can show that they behave alike on the mathematically interesting classes chosen for investigation in [3]. We give a “Pull-back Theorem”, saying that if Ф is a Turing computable embedding of K into K′ , then for any computable infinitary sentence φ in the language of K′, we can find a computable infinitary sentence φ* in the language of K such that for all A ∈ K A ⊨ φ* iff Φ ( A ) ⊨ φ and φ* has the same “complexity” as φ (i.e., if φ is computable Σ α or computable Π α , for α ≥ 1, then so is φ*). The Pull-back Theorem is useful in proving non-embeddability, and it has other applications as well.
Abstract The first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelian p -groups, the class of Abelian torsion groups, …
Abstract The first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelian p -groups, the class of Abelian torsion groups, and the special class of “rank-homogeneous” trees. We consider these conditions as a possible definition of what it means for a class of structures to have “Ulm type”. The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply this result to show that there is no Turing computable embedding of the class of graphs into the class of “rank-homogeneous” trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a “rank-preserving” Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank .
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 …
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 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” …
Abstract 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.
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 …
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 reader the background material needed to understand almost any paper on the subject. The second is to advertise the use of honest definitions, in particular in establishing basic results, such as the so-called shrinking of indiscernibles.
Abstract Fix a language extending the language of ordered fields by at least one new predicate or function symbol. Call an L -structure R pseudo-o-minimal if it is (elementarily equivalent …
Abstract Fix a language extending the language of ordered fields by at least one new predicate or function symbol. Call an L -structure R pseudo-o-minimal if it is (elementarily equivalent to) an ultraproduct of o-minimal structures. We show that for any recursive list of L -sentences , there is a real closed field satisfying which is not pseudo-o-minimal. This shows that the theory T o−min consisting of those -sentences true in all o-minimal -structures, also called the theory of o-minimality (for L) , is not recursively axiomatizable. And, in particular, there are locally o-minimal, definably complete expansions of real closed fields which are not pseudo-o-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 …
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.
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, …
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.
This paper provides an overview of recent work by the authors and others on two topics in the model theory of finite structures. The point of view here differs from …
This paper provides an overview of recent work by the authors and others on two topics in the model theory of finite structures. The point of view here differs from that usually associated with the term 'finite model theory', as presented for example in [21] or [46], in which the emphasis and motivation come primarily from computer science. Instead, the inspiration for this work has its origins in contemporary (infinite) model theoretic themes such as dimension, independence, and various measures of the complexity of definable sets. Each of the topics deals with classes of finite structures for first-order logic that are isolated by conditions that are drawn from these model-theoretic considerations. Moreover, in both cases, connections exist to areas in infinite model theory such as stability and simplicity theory, and o-minimality. This survey is intended for both mathematical logicians and computer scientists whose work focuses on logical aspects of the subject.
In this paper, we study Vapnik‐Chervonenkis density (VC‐density) over indiscernible sequences (denoted VC ind ‐density). We answer an open question in [1], showing that VC ind ‐density is always integer …
In this paper, we study Vapnik‐Chervonenkis density (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.
We will study complete L n -theories and their models, where L n is the set of first order formulas in which at most n distinct variables occur. Here, by …
We will study complete L n -theories and their models, where L n is the set of first order formulas in which at most n distinct variables occur. Here, by a complete L n -theory we mean a theory such that for every L n -sentence, it or its negation is implied by the theory. Hence, a complete L n -theory need not necessarily be complete in the usual sense. Our approach is to transfer concepts and methods from stability theory, such as the order property and counting types, to the context of L n -theories. So, in one sense, we will develop some rudimentary stability theory for a particular class of (possibly) incomplete theories. To make the ‘stability theoretic’ arguments work, we need to assume that models of the complete L n -theory T which we consider can be amalgamated in certain ways. If this condition is satisfied and T has infinite models then there will exist models of T which are sufficiently saturated with respect to L n . This allows us to use some counting types arguments from stability theory. If, moreover, we impose some finiteness conditions on the number of L n -types and the length of L n -definable orders then a sufficiently saturated model of T will be ω -categorical and ω -stable. Using the theory of ω -categorical and ω -stable structures we derive that T has arbitrarily large finite models. A different approach to combining stability theory with finite model theory is made by Hyttinen in [9] and [10].
Abstract We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF -reducibility introduced in [9] to show completeness of the isomorphism …
Abstract We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF -reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all equivalence relations on hyperarithmetical subsets of ω .