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Additivity of the dprank
"... The main result is the prove of the linearity of the dprank. We also prove that the study of theories of finite dprank cannot be reduced to the study of its dpminimal types and discuss the possible relations between dprank and VCdensity. ..."
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The main result is the prove of the linearity of the dprank. We also prove that the study of theories of finite dprank cannot be reduced to the study of its dpminimal types and discuss the possible relations between dprank and VCdensity.
EDGE DISTRIBUTION AND DENSITY IN THE CHARACTERISTIC SEQUENCE
, 909
"... Abstract. The characteristic sequence of hypergraphs 〈Pn: n < ω 〉 associated to a formula ϕ(x; y), introduced in [6], is defined by Pn(y1,... yn) = (∃x) ∧ i≤n ϕ(x; yi). This paper continues the study of characteristic sequences, showing that graphtheoretic techniques, notably Szemerédi’s celebr ..."
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Abstract. The characteristic sequence of hypergraphs 〈Pn: n < ω 〉 associated to a formula ϕ(x; y), introduced in [6], is defined by Pn(y1,... yn) = (∃x) ∧ i≤n ϕ(x; yi). This paper continues the study of characteristic sequences, showing that graphtheoretic techniques, notably Szemerédi’s celebrated regularity lemma, can be naturally applied to the study of modeltheoretic complexity via the characteristic sequence. Specifically, we relate classificationtheoretic properties of ϕ and of the Pn (considered as formulas) to density between components in Szemerédiregular decompositions of graphs in the characteristic sequence. In addition, we use Szemerédi regularity to calibrate modeltheoretic notions of independence by describing the depth of independence of a constellation of sets and showing that certain failures of depth imply Shelah’s strong order property SOP3; this sheds light on the interplay of independence and order in unstable theories. 1.
Generically stable and smooth measures in NIP theories
, 2010
"... We formulate the measure analogue of generically stable types in first order theories with NIP (without the independence property), giving several characterizations, answering some questions from [9], and giving another treatment of uniqueness results from [9]. We introduce a notion of “generic comp ..."
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We formulate the measure analogue of generically stable types in first order theories with NIP (without the independence property), giving several characterizations, answering some questions from [9], and giving another treatment of uniqueness results from [9]. We introduce a notion of “generic compact domination”, relating it to stationarity of Keisler measures, and also giving group versions. We also prove the “approximate definability ” of arbitrary Borel probability measures on definable sets in the real and padic fields. 1 Introduction and
ORTHOGONALITY AND DOMINATION IN UNSTABLE THEORIES
"... Abstract. In the rst part of the paper we study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and superrosy theories. Then we try to develop analogous theory for arbitrary dependent theories. 1. Introduction and ..."
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Abstract. In the rst part of the paper we study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and superrosy theories. Then we try to develop analogous theory for arbitrary dependent theories. 1. Introduction and
PERSISTENCE AND NIP IN THE CHARACTERISTIC SEQUENCE
, 908
"... Abstract. For a firstorder formula ϕ(x; y) we introduce and study the characteristic sequence 〈Pn: n < ω 〉 of hypergraphs defined by Pn(y1,...,yn): = (∃x) ∧ i≤n ϕ(x; yi). We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theo ..."
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Abstract. For a firstorder formula ϕ(x; y) we introduce and study the characteristic sequence 〈Pn: n < ω 〉 of hypergraphs defined by Pn(y1,...,yn): = (∃x) ∧ i≤n ϕ(x; yi). We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theoretic properties of ϕ and vice versa. 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. 1.
Ramakrishnan, Janak Daniel/Research Statement/20071210
"... unstable theories. A theory is unstable if there is a formula that orders an infinite set of tuples in some model of the theory. A theory is independent if there is a definable binary relation on tuples and, in sufficiently saturated models, an infinite set, such that every subset of this set is pic ..."
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unstable theories. A theory is unstable if there is a formula that orders an infinite set of tuples in some model of the theory. A theory is independent if there is a definable binary relation on tuples and, in sufficiently saturated models, an infinite set, such that every subset of this set is picked out by the relation and some tuple in the model. Dependent theories are those without this property, and dependent (unstable) structures are those with a dependent (unstable) theory. Unstable dependent structures always have definable partial orders with arbitrarily long chains. Ominimal structures are linearly ordered structures in which every definable subset is a finite union of points and intervals, and can be considered the “tamest” examples of dependent unstable theories. In ominimality, my research has focused on classification of types in ominimal theories, as first proposed by [Mar86] and extended by [Dol04]. There is a simple dichotomy of Dedekind cuts in a linear order: when both sides of the cut are open sets, denoted cuts, and when at least one side is closed, denoted noncuts. Since the order type induces the type in ominimal theories, this classification proves quite