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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)

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

- 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)

- 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)

- 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 "… (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
- 2016

- 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)

- 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)

- 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)