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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… (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 ≤… (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… (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)… (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… (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… (More)

We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler’s order. Thus Keisler’s order is infinite and not a well order. Moreover, this chain occurs… (More)

- Noga Alon, Roi Livni, Maryanthe Malliaris, Shay Moran
- STOC
- 2019

We show that every approximately differentially private learning algorithm (possibly improper) for a class H with Littlestone dimension d requires Ω(log*(d)) examples. As a corollary it follows that… (More)

Persistence and Regularity in Unstable Model Theory

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

Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers… (More)