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**11 - 19**of**19**### PERSISTENCE AND NIP IN THE CHARACTERISTIC SEQUENCE

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"... Abstract. For a first-order 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 first-order 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.

### THESIS WORK: “PERSISTENCE AND REGULARITY IN UNSTABLE MODEL THEORY”

"... Historically one of the great successes of model theory has been Shelah’s stability theory: a program, described in [17], of showing that the arrangement of first-order theories into complexity classes according to a priori set-theoretic criteria (e.g. counting types over sets) in fact pushes down t ..."

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Historically one of the great successes of model theory has been Shelah’s stability theory: a program, described in [17], of showing that the arrangement of first-order theories into complexity classes according to a priori set-theoretic criteria (e.g. counting types over sets) in fact pushes down to reveal a very rich and entirely model-theoretic structure theory for the classes involved: what we now call stability, superstability, and ω-stability, as well as the dichotomy between independence and strict order in unstable theories. The success of the program may be measured by the fact that the original set-theoretic criteria are now largely passed over in favor of definitions which mention ranks or combinatorial properties of a particular formula. Because of this shift, Keisler’s 1967 order (defined below) may strike the modern reader as an anachronism. It too seeks to coarsely classify first-order theories in terms of a more set-theoretic criterion, the difficulty of producing saturated regular ultrapowers, but its structure has remained largely open. Partial results from the 70s suggest a mine of perhaps comparable richness, one which has remained largely inaccessible to current tools. Keisler’s criterion of choice, saturation of regular ultrapowers, is natural for two reasons. First, when the ultrapower is regular, the degree of its saturation depends only on the theory and not on the saturation of the index models. Second, ultrapowers are a natural context for studying compactness, and Keisler’s order can be thought of as studying the fine structure of compactness by asking: what families of consistent types are realized or omitted together in regular ultrapowers? Thus the relative difficulty of realizing the types of T1 versus those of some T2 in regular ultrapowers gives a measure of the combinatorial complexity of the types each Ti is able to describe. Definition 1. (Keisler’s order [7]) T1 ≤ T2 if for all infinite λ, D regular on λ, M1 |= T1,M2 | = T2, we have: if (M2)λ/D is λ+-saturated then (M1)λ/D is λ+-saturated.

### EXISTENCE OF OPTIMAL ULTRAFILTERS AND THE FUNDAMENTAL COMPLEXITY OF SIMPLE THEORIES

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### THE METAMATHEMATICS OF RANDOM GRAPHS

"... We explain and summarize the use of logic to provide a uniform perspective for studying limit laws on finite probability spaces. This work model theory, and probability. We conclude by linking this context with work on the Urysohn space. Erdös pioneered the use of the probabilistic method for provin ..."

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We explain and summarize the use of logic to provide a uniform perspective for studying limit laws on finite probability spaces. This work model theory, and probability. We conclude by linking this context with work on the Urysohn space. Erdös pioneered the use of the probabilistic method for proving statements in finite combinatorics. In this paper we explain how logic is used to formalize and give general proofs for a large class of such arguments. We consider the role of the logic, the probability measure, and the vocabulary in formulating the problems. We report a number of results in this area, spotlighting the Baldwin-Shelah method of determined theories. In this paper we explore some of the surprising connections between diverse areas which appeared at this conference. On the one hand we discuss the use of a specific Abstract Elementary Class as a tool for proving 0-1 laws. On the other, we conclude with a formulation of issues relating to the Urysohn space in the framework for studying random graphs developed here. Here is a specific example of the use of the probabilistic method. A (round robin) tournament is a directed graph with an edge between every pair of points. Fix k. Is there a tournament that satisfies Pk: for each set of k-players there is another who beats each of them? Here is a method for showing the answer is yes. Let Sn be the set of all tournaments with n players. Then, |Sn | = 2 (n2). Each of these is equally likely. Call a k-set X bad if no element dominates each member of X. If Y (T) is the number of bad k-sets in a tournament T then

### INDEPENDENCE, ORDER, AND THE INTERACTION OF ULTRAFILTERS AND THEORIES

"... Abstract. We consider the question, of longstanding interest, of realizing types in regular ultra-powers. In particular, this is a question about the interaction of ultrafilters and theories, which is both coarse and subtle. By our prior work it suffices to consider types given by instances of a sin ..."

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Abstract. We consider the question, of longstanding interest, of realizing types in regular ultra-powers. In particular, this is a question about the interaction of ultrafilters and theories, which is both coarse and subtle. By our prior work it suffices to consider types given by instances of a single formula. In this article, we analyze a class of formulas ϕ whose associated characteristic sequence of hypergraphs can be seen as describing realization of first- and second-order types in ultrapowers on one hand, and properties of the corresponding ultrafilters on the other. These formulas act, via the characteristic sequence, as points of contact with the ultrafilter D, in the sense that they translate structural properties of ultrafilters into model-theoretically meaningful properties and vice versa. Such formulas characterize saturation for various key theories (e.g. Trg, Tfeq), yet their scope in Keisler’s order does not extend beyond Tfeq. The proof applies Shelah’s classification of second-order quantifiers. 1.

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"... The starting point is a question about the structure of Keisler’s order, a preorder on theories which compares the difficulty of producing saturated regular ultrapow-ers. In Chapter 1 we show that Keisler’s order reduces to the analysis of types in a finite language, i.e. that the combinatorial barr ..."

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The starting point is a question about the structure of Keisler’s order, a preorder on theories which compares the difficulty of producing saturated regular ultrapow-ers. In Chapter 1 we show that Keisler’s order reduces to the analysis of types in a finite language, i.e. that the combinatorial barriers to saturation are contained in the parameter spaces of the formulas of T. In Chapter 2 we define the character-istic sequence of hypergraphs 〈Pn: n < ω 〉 associated to a formula which describe the relevant incidence relations, and develop a general framework for analyzing the complexity of a formula in terms of the complexity of its characteristic sequence. Specifically, we are interested in analyzing consistent partial types, which corre-spond to sets A such that An ⊂ Pn for all n. The key issues studied in Chapter 2 are localization and persistence, which describe the difficulty of separating some 2fixed complex configuration from a complete graph under analysis by progressive re-strictions of the base set. We characterize stability and simplicity of ϕ in terms of persistence in the characteristic sequence. Chapter 3 restricts attention to the behavior of the graph P2 in the character-

### Torsion-free hyperbolic groups and the finite cover property

, 2013

"... We prove that the first order theory of non abelian free groups eliminates the ∃∞-quantifier (in eq). Equivalently, since the theory of non abelian free groups is stable, it does not have the finite cover property. We extend our results to torsion-free hyperbolic groups under some conditions. 1 ..."

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We prove that the first order theory of non abelian free groups eliminates the ∃∞-quantifier (in eq). Equivalently, since the theory of non abelian free groups is stable, it does not have the finite cover property. We extend our results to torsion-free hyperbolic groups under some conditions. 1