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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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Cited by 12 (11 self)
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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, arXiv eprint 1208.5424
, 2012
"... Abstract. We connect and solve two longstanding open problems in quite different areas: the modeltheoretic question of whether SOP2 is maximal in Keisler’s order, and the question from general topology/set theory of whether p = t, the oldest problem on cardinal invariants of the continuum. We do so ..."
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Cited by 8 (5 self)
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Abstract. We connect and solve two longstanding open problems in quite different areas: the modeltheoretic question of whether SOP2 is maximal in Keisler’s order, and the question from general topology/set theory of whether p = t, the oldest problem on cardinal invariants of the continuum. We do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely, using modeltheoretic methods. By a cofinality spectrum problem s we essentially mean the data of a pair of models M M1 which code sufficient set theory, possibly in an expanded language, along with a distinguished set of formulas ∆s which define linear orders in M1. Let ts, the “treetops ” of s, be the smallest regular cardinal λ such that one of a set of derived trees in M1 has a strictly increasing λsequence with no upper bound. Let C(s, ts) be the set of pairs of regular cardinals (κ1, κ2) such that κ1 ≤ κ2 < ts and some ∆sdefinable linear order contains a (κ1, κ2)cut. We prove that for any cofinality spectrum problem s, C(s, ts) = ∅. Using this theorem and framework we prove first, that SOP2 is maximal in Keisler’s order; second, that p = t; and third, that any regular ultrafilter D on λ for which “ts> λ, ” or what is equivalent, such that (ω,<)λ/D contains no (κ, κ)cuts for κ = cf(κ) ≤ λ, is λ+good. We obtain several consequences, notably existence of a minimum Keisler class among the nonsimple theories.
A ring homomorphism is enough to get nonstandard analysis
"... Abstract. It is shown that assuming the existence of a suitable ring homomorphism is enough to get an algebraic presentation of nonstandard methods that is equivalent to the popular superstructure approach, including κsaturation. 1. ..."
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Cited by 5 (3 self)
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Abstract. It is shown that assuming the existence of a suitable ring homomorphism is enough to get an algebraic presentation of nonstandard methods that is equivalent to the popular superstructure approach, including κsaturation. 1.
Saturating the random graph with an independent family of small range
, 2012
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EXISTENCE OF OPTIMAL ULTRAFILTERS AND THE FUNDAMENTAL COMPLEXITY OF SIMPLE THEORIES
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REGULAR MAPPINGS OF SEQUENCE SPACES OVER FINITE FIELDS
"... Let K be a finite field and let X be the set of doubly infinite sequences over K, that is, X = K z where Z denotes the set of integers. In this paper I shall study some problems about mappings from X to itself which are related to earlier results in topological dynamics and cryptography. The second ..."
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Let K be a finite field and let X be the set of doubly infinite sequences over K, that is, X = K z where Z denotes the set of integers. In this paper I shall study some problems about mappings from X to itself which are related to earlier results in topological dynamics and cryptography. The second section includes a
INDEPENDENCE, ORDER, AND THE INTERACTION OF ULTRAFILTERS AND THEORIES
"... Abstract. We consider the question, of longstanding interest, of realizing types in regular ultrapowers. 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 ultrapowers. 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 secondorder 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 modeltheoretically 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 secondorder quantifiers. 1.