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- M. MALLIARIS
- 2013

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)

- M. MALLIARIS
- 2013

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

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 "… (More)

- M. MALLIARIS
- 2012

We connect and solve two longstanding open problems in quite different areas: the model-theoretic question of whether SOP 2 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)

Historically one of the great successes of model theory has been Shelah's stability theory: a program, described in [17], of showing that the arrangement of first-order theories into complexity classes according to a priori set-theoretic criteria (e.g. counting types over sets) in fact pushes down to reveal a very rich and entirely model-theoretic structure… (More)

- M. MALLIARIS
- 2011

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)

- M. Malliaris, S. Shelah
- 2015

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)

We prove a regularity lemma with respect to arbitrary Keisler measures µ on V , ν on W where the bipartite graph (V, W, R) is definable in a saturated structure ¯ M and the formula R(x, y) is stable. The proof is rather quick, making use of local stability theory. The special case where (V, W, R) is pseudofinite, µ, ν are the counting measures, and ¯ M is… (More)

The first part of this paper is an expository overview of the au-thors' 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)

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