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We develop a framework in which Szemerédi’s celebrated Regularity Lemma for graphs interacts with core model-theoretic ideas and techniques. Our work relies on a coincidence of ideas from model theory and graph theory: arbitrarily large half-graphs coincide with model-theoretic instability, so in their absence, structure theorems and technology from… (More)

We connect and solve two longstanding open problems in quite different areas: the model-theoretic 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… (More)

This paper contributes to the set-theoretic side of understanding Keisler’s order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality lcf(א0,D) of א0 modulo D, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts.… (More)

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, we construct a regular ultrafilter D on λ such that (i) for any model M of a stable theory or of the random graph, M/D is λ-saturated but (ii) if Th(N) is… (More)

- Ramin Takloo-Bighash, Henri Gillet, +7 authors Phyllis J. Cassidy
- 2012

- Maryanthe Malliaris, Saharon Shelah
- Proceedings of the National Academy of Sciences…
- 2013

Cantor proved in 1874 [Cantor G (1874) J Reine Angew Math 77:258-262] that the continuum is uncountable, and Hilbert's first problem asks whether it is the smallest uncountable cardinal. A program arose to study cardinal invariants of the continuum, which measure the size of the continuum in various ways. By Gödel [Gödel K (1939) Proc Natl Acad Sci USA… (More)

In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the… (More)

- Maryanthe Malliaris, Saharon Shelah
- J. Symb. Log.
- 2014

Via two short proofs and three constructions, we show how to increase the model-theoretic 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… (More)

- Maryanthe Malliaris, Saharon Shelah
- Logic Without Borders
- 2015

The first part of this paper is an expository overview of the authors’ recent work on Keisler’s order, a far-reaching program of understanding basic model-theoretic structure through the lens of regular ultrapowers. We motivate the problem and explain how this work connects model theory and set theory, leading to theorems on both sides. In the second part… (More)

Chudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs [1]. In the present paper we reprove their theorem using many of the same ideas, but with the key model-theoretic ingredient of first determining the so-called amount of stability of the graph. This approach changes the… (More)