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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 10 (9 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.
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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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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DOI: 10.1112/S0024609304003431 A NEW WEIL COHOMOLOGY
"... There is a Weil cohomology theory with coefficients in a pseudofinite field of characteristic zero. 1. ..."
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There is a Weil cohomology theory with coefficients in a pseudofinite field of characteristic zero. 1.
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
EXISTENCE OF OPTIMAL ULTRAFILTERS AND THE FUNDAMENTAL COMPLEXITY OF SIMPLE THEORIES
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