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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.” math.LO/1208.2579
"... Abstract. 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 satura ..."
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Abstract. 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. 1.
Saturating the random graph with an independent family of small range
, 2012
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Version of 1.7.14 p = t, following MalliarisShelah and Steprāns D.H.Fremlin
"... I attempt a proof, based on that sketched in Steprāns n13, of the theorem in Malliaris & Shelah p13 that p = t. 1 Gaps, interpolation and chainadditivity 1A Definitions Let P be a partially ordered set and λ, κ nonzero cardinals. (a) A (λ, κ∗)gap in P is a pair (〈xξ〉ξ<λ, 〈yξ〉η<κ) of fa ..."
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I attempt a proof, based on that sketched in Steprāns n13, of the theorem in Malliaris & Shelah p13 that p = t. 1 Gaps, interpolation and chainadditivity 1A Definitions Let P be a partially ordered set and λ, κ nonzero cardinals. (a) A (λ, κ∗)gap in P is a pair (〈xξ〉ξ<λ, 〈yξ〉η<κ) of families in P such that xξ < xξ ′ ≤ yη ′ < yη whenever ξ < ξ ′ < λ and η < η ′ < κ, there is no z ∈ P such that xξ ≤ z ≤ yη whenever ξ < λ and η < κ. (a) A peculiar (λ, κ∗)gap in P is a pair (〈xξ〉ξ<λ, 〈yξ〉η<κ) of families in P such that xξ < xξ ′ ≤ yη ′ < yη whenever ξ < ξ ′ < λ and η < η ′ < κ, whenever z ∈ P is such that z ≤ yη for every η < κ, there is a ξ < λ such that z ≤ xξ, whenever z ∈ P is such that xξ ≤ z for every ξ < λ, there is an η < κ such that yη ≤ z. 1B Definitions Let (P,≤) be a partially ordered set. (a) The chainadditivity of P, chaddP, is the least cardinal of any totally ordered subset of P with no