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Hypergraph sequences as a tool for saturation of ultrapowers
 Journal of Symbolic Logic
, 2012
"... Abstract. Let T1, T2 be countable firstorder theories, Mi  = Ti, and D any regular ultrafilter on λ ≥ ℵ0. A longstanding open problem of Keisler asks when T2 is more complex than T1, as measured by the fact that for any such λ,D, if the ultrapower (M2)λ/D realizes all types over sets of size ≤ λ, ..."
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Abstract. Let T1, T2 be countable firstorder theories, Mi  = Ti, and D any regular ultrafilter on λ ≥ ℵ0. A longstanding open problem of Keisler asks when T2 is more complex than T1, as measured by the fact that for any such λ,D, if the ultrapower (M2)λ/D realizes all types over sets of size ≤ λ, then so must the ultrapower (M1)λ/D. In this paper, building on the author’s prior work [11] [12] [13], we show that the relative complexity of firstorder theories in Keisler’s sense is reflected in the relative graphtheoretic complexity of sequences of hypergraphs associated to formulas of the theory. After reviewing prior work on Keisler’s order, we present the new construction in the context of ultrapowers, give various applications to the open question of the unstable classification, and investigate the interaction between theories and regularizing sets. We show that there is a minimal unstable theory, a minimal TP2 theory, and that maximality is implied by the density of certain graph edges (between components arising from Szemerédiregular decompositions) remaining bounded away from 0, 1. We also introduce and discuss flexible ultrafilters, a relevant class of regular ultrafilters which reflect the sensitivity of certain unstable (non low) theories to the sizes of regularizing sets, and prove that any ultrafilter which saturates the minimal TP2 theory is flexible.
Cofinality spectrum theorems in model theory, set theory and general topology
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CONSTRUCTING REGULAR ULTRAFILTERS FROM A MODELTHEORETIC POINT OF VIEW
"... Abstract. This paper contributes to the settheoretic side of understanding Keisler’s order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality lcf(ℵ0,D) of ℵ0 modulo D, saturation of the minimum unstable theory (the random graph), flexibility, ..."
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Abstract. This paper contributes to the settheoretic side of understanding Keisler’s order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality lcf(ℵ0,D) of ℵ0 modulo D, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, known to be detected by nonlow theories. Assuming κ> ℵ0 is measurable, we construct a regular ultrafilter on λ ≥ 2 κ which is flexible but not good, and which moreover has large lcf(ℵ0) but does not even saturate models of the random graph. This implies (a) that flexibility alone cannot characterize saturation of any theory, however (b) by separating flexibility from goodness, we remove a main obstacle to proving nonlow does not imply maximal. Since flexible is precisely OK, this also shows that (c) from a settheoretic point of view, consistently, ok need not imply good, addressing a problem from Dow 1985. Second, under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and also give a new proof that there is a loss of saturation in ultrapowers of nonsimple theories. More precisely, for D regular on κ and M a
A dividing line within simple unstable theories.
, 2013
"... We give the first (ZFC) dividing line in Keisler’s order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal λ for which there is µ < λ ≤ 2 µ, we construct a regular ultrafilter D on λ so that (i) for any model M of a stable theory or ..."
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We give the first (ZFC) dividing line in Keisler’s order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal λ for which there is µ < λ ≤ 2 µ, we construct a regular ultrafilter D on λ so that (i) for any model M of a stable theory or of the random graph, M λ /D is λ +saturated but (ii) if Th(N) is not simple or not low then N λ /D is not λ +saturated. The nonsaturation result relies on the notion of flexible ultrafilters. To prove the saturation result we develop a property of a class of simple theories, called Qr 1, generalizing the fact that whenever B is a set of parameters in some sufficiently saturated model of the random graph, B  = λ and µ < λ ≤ 2 µ, then there is a set A with A  = µ so that any nonalgebraic p ∈ S(B) is finitely realized in A. In addition to giving information about simple unstable theories, our proof reframes the problem of saturation of ultrapowers in several key ways. We give a new characterization of good filters in terms of “excellence, ” a measure of the accuracy of the quotient Boolean algebra. We introduce and develop the notion of moral ultrafilters on Boolean algebras. We prove a socalled “separation of variables ” result which shows how the problem of constructing ultrafilters to have a precise degree of saturation may be profitably separated into a more settheoretic stage, building an excellent filter, followed by a more modeltheoretic stage: building socalled moral ultrafilters on the quotient Boolean algebra, a process which highlights the complexity of certain patterns, arising from firstorder formulas, in certain Boolean algebras.
Modeltheoretic properties of ultrafilters built by independent families of functions
"... Via two short proofs and three constructions, we show how to increase the modeltheoretic precision of a widely used method for building ultrafilters. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any ..."
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Via two short proofs and three constructions, we show how to increase the modeltheoretic precision of a widely used method for building ultrafilters. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck ” in the inductive construction of a regular ultrafilter on λ (i.e. a point after which all antichains of P(λ)/D have cardinality less than λ) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any nonlow theory. The constructions are as follows. First, we construct a regular filter D on λ so that any ultrafilter extending D fails to λ+saturate ultrapowers of the random graph, thus of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal κ, we construct a regular ultrafilter on λ> κ which is λflexible but not κ++good, improving our previous answer to a question raised in Dow 1975. Third, assuming a weakly compact cardinal κ, we construct an ultrafilter to show that lcf(ℵ0) may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of precuts may be realized while still failing to saturate any unstable theory. Our methods advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation.
Edge distribution and density in the characteristic sequence
, 2009
"... 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 regu ..."
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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.
Transferring saturation, the finite cover property and stability
 Journal of Symbolic Logic
, 1999
"... Saturation is (µ, κ)transferable in T if and only if there is an expansion T1 of T with T1  = T  such that if M is a µsaturated model of T1 and M  ≥ κ then the reduct ML(T) is κsaturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively th ..."
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Saturation is (µ, κ)transferable in T if and only if there is an expansion T1 of T with T1  = T  such that if M is a µsaturated model of T1 and M  ≥ κ then the reduct ML(T) is κsaturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (ℵ0, λ)transferable or (κ(T), λ)transferable for all λ. Further if for some µ ≥ T , 2 µ> µ +, stability is equivalent to for all µ ≥ T , saturation is (µ, 2 µ)transferable. 1
Finite models, stability, and Ramsey’s theorem
, 2008
"... We prove some results on the border of Ramsey theory (finite partition calculus) and model theory. Also a beginning of classification theory for classes of finite models is undertaken. ..."
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We prove some results on the border of Ramsey theory (finite partition calculus) and model theory. Also a beginning of classification theory for classes of finite models is undertaken.
Saturating the random graph with an independent family of small range
, 2012
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