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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.
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.