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Hypergraph sequences as a tool for saturation of ultrapowers

  • M. E. Malliaris (a1)

Let T1, T2 be countable first-order theories, MiTi and 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, , if the ultrapower realizes all types over sets of size ≤ λ, then so must the ultrapower . In this paper, building on the author's prior work [12] [13] [14], we show that the relative complexity of first-order theories in Keisler's sense is reflected in the relative graph-theoretic 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 minimum unstable theory, a minimum TP2 theory, and that maximality is implied by the density of certain graph edges (between components arising from Szemerédi-regular 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.

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[1] J. Baker and K. Kunen , Limits in the uniform ultrafilters, Transactions of the American Mathematical Society, vol. 353 (2001), no. 10, pp. 40834093.

[3] W. Comfort and S. Negrepontis , The theory of ultrafilters, Springer-Verlag, 1974.

[4] A. Dow , Good and ok ultrafilters, Transactions of the American Mathematical Society, vol. 290 (1985), no. 1, pp. 145160.

[5] M. Džamonja and S. Shelah , On ⊲*-maximality, Annals of Pure and Applied Logic, vol. 125 (2004), pp. 119158.

[9] S. Kochen , Ultraproducts in the theory of models, Annals of Mathematics. Second Series, vol. 74 (1961), no. 2, pp. 221261.

[11] K. Kunen , Ultrafilters and independent sets, Transactions of the American Mathematical Society, vol. 172 (1972), pp. 299306.

[12] M. Malliaris , Realization of φ-types and Keisler's order, Annals of Pure and Applied Logic, vol. 157 (2009), pp. 220224.

[14] M. Malliaris , Edge distribution and density in the characteristic sequence, Annals of Pure and Applied Logic, vol. 162 (2010), no. 1, pp. 119.

[15] S. Shelah , Simple unstable theories, Annals of Mathematical Logic, vol. 19 (1980), pp. 177203.

[18] S. Shelah , Toward classifying unstable theories, Annals of Pure and Applied Logic, vol. 80 (1996), pp. 229255.

[19] S. Shelah and A. Usvyatsov , More on SOP1and SOP2, Annals of Pure and Applied Logic, vol. 155 (2008), no. 1, pp. 1631.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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