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M. Malliaris

Publications and source records attributed to M. Malliaris.

At least 19 recordsLinked to original sources
arXiv2023

Shearing in some simple rank one theories

Dividing asks about inconsistency along indiscernible sequences. In order to study the finer structure of simple theories without much dividing, the authors recently introduced shearing, which essentially asks about inconsistency along generalized indiscernible sequences. Here we characterize the shearing of the random graph. We then use shearing to distinguish between the random graph and the theories $T_{n,k}$, the higher-order analogues of the triangle-free random graph. It follows that shearing is distinct from dividing in simple unstable theories, and distinguishes meaningfully between classes of simple unstable rank one theories. The paper begins with an overview of shearing, and includes open questions.

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arXiv2023

Some simple theories from a Boolean algebra point of view

We find a strong separation between two natural families of simple rank one theories in Keisler's order: the theories $T_\mathfrak{m}$ reflecting graph sequences, which witness that Keisler's order has the maximum number of classes, and the theories $T_{n,k}$, which are the higher-order analogues of the triangle-free random graph. The proof involves building Boolean algebras and ultrafilters "by hand" to satisfy certain model theoretically meaningful chain conditions. This may be seen as advancing a line of work going back through Kunen's construction of good ultrafilters in ZFC using families of independent functions. We conclude with a theorem on flexible ultrafilters, and open questions.

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arXiv2021

New simple theories from hypergraph sequences

We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler's order has continuum many classes, however, the construction does not require familiarity with the earlier proof.) We prove a model-completion and quantifier-elimination result for theories in this family. We develop a combinatorial property which they share. We invoke regular ultrafilters to show the strength of this property, showing that any flexible ultrafilter which is good for the random graph is able to saturate such theories.

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arXiv2021

Keisler's order is not simple (and simple theories may not be either)

Solving a decades-old problem we show that Keisler's 1967 order on theories has the maximum number of classes. The theories we build are simple unstable with no nontrivial forking, and reflect growth rates of sequences which may be thought of as densities of certain regular pairs, in the sense of Szemerédi's regularity lemma. The proof involves ideas from model theory, set theory, and finite combinatorics.

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arXiv2021

A separation theorem for simple theories

This paper builds model-theoretic tools to detect changes in complexity among the simple theories. We develop a generalization of dividing, called shearing, which depends on a so-called context c. This leads to defining c-superstability, a syntactical notion, which includes supersimplicity as a special case. We prove a separation theorem showing that for any countable context c and any two theories $T_1$, $T_2$ such that $T_1$ is c-superstable and $T_2$ is c-unsuperstable, and for arbitrarily large $μ$, it is possible to build models of any theory interpreting both $T_1$ and $T_2$ whose restriction to $τ(T_1)$ is $μ$-saturated and whose restriction to $τ(T_2)$ is not $\aleph_1$-saturated. (This suggests "c-superstable" is really a dividing line.) The proof uses generalized Ehrenfeucht-Mostowski models, and along the way, we clarify the use of these techniques to realize certain types while omitting others. In some sense, shearing allows us to study the interaction of complexity coming from the usual notion of dividing in simple theories and the more combinatorial complexity detected by the general definition. This work is inspired by our recent progress on Keisler's order, but does not use ultrafilters, rather aiming to build up the internal model theory of these classes.

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arXiv2021

Notes on the stable regularity lemma

This is a short expository account of the regularity lemma for stable graphs proved by the authors, with some comments on the model theoretic context, written for a general logical audience.

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arXiv2019

An example of a new simple theory

We construct a countable simple theory which, in Keisler's order, is strictly above the random graph (but "barely so") and also in some sense orthogonal to the building blocks of the recently discovered infinite descending chain. As a result we prove in ZFC that there are incomparable classes in Keisler's order.

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arXiv2019

A new look at interpretability and saturation

We investigate the interpretability ordering $\trianglelefteq^*$ using generalized Ehrenfeucht-Mostowski models. This gives a new approach to proving inequalities and investigating the structure of types.

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arXiv2015

On unavoidable induced subgraphs in large prime graphs

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 applicable Ramsey theorem, improves the bounds and offers a different structural perspective on the graphs in question. Complementing this, we give an infinitary proof which implies the finite result.

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arXiv2015

Keisler's order has infinitely many classes

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 within the simple unstable theories, considered model-theoretically tame. Keisler's order is a central notion of the model theory of the 60s and 70s which compares first-order theories (and implicitly ultrafilters) according to saturation of ultrapowers. Prior to this paper, it was long thought to have finitely many classes, linearly ordered. The model-theoretic complexity we find is witnessed by a very natural class of theories, the $n$-free $k$-hypergraphs studied by Hrushovski. This complexity reflects the difficulty of amalgamation and appears orthogonal to forking.

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arXiv2015

Existence of optimal ultrafilters and the fundamental complexity of simple theories

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 global theory, and introducing nonforking, seminal for the development of stability theory. Prior to the present paper, there had been no such characterization of an unstable class. In the present paper, we first establish the existence of so-called optimal ultrafilters on Boolean algebras, which are to simple theories as Keisler's good ultrafilters are to all theories. Then, assuming a supercompact cardinal, we characterize the simple theories in terms of saturation of ultrapowers. To do so, we lay the groundwork for analyzing the global structure of simple theories, in ZFC, via complexity of certain amalgamation patterns. This brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability.

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arXiv2015

Cofinality spectrum theorems in model theory, set theory and general topology

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 set theory/general topology of whether $\mathfrak{p} = \mathfrak{t}$, the oldest problem on cardinal invariants of the continuum. We do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely, using model-theoretic methods.

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arXiv2015

Model-theoretic applications of cofinality spectrum problems

We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is $λ$-saturated iff it has cofinality $\geq λ$ and the underlying order has no $(κ, κ)$-cuts for regular $κ< λ$. Second, assuming instances of GCH, we prove that $SOP_2$ characterizes maximality in the interpretability order $\trianglelefteq^*$, settling a prior conjecture and proving that $SOP_2$ is a real dividing line. Third, we establish the beginnings of a structure theory for $NSOP_2$, proving that $NSOP_2$ can be characterized by the existence of few inconsistent higher formulas. In the course of the paper, we show that $\mathfrak{p}_s = \mathfrak{t}_s$ in any weak cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in cofinality spectrum problems arising from Peano arithmetic.

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arXiv2012

Saturating the random graph with an independent family of small range

Motivated by Keisler's order, a far-reaching program of understanding basic model-theoretic structure through the lens of regular ultrapowers, we prove that for a class of regular filters $D$ on $I$, $|I| = λ> \aleph_0$, the fact that $P(I)/\de$ has little freedom (as measured by the fact that any maximal antichain is of size $<λ$, or even countable) does not prevent extending $D$ to an ultrafilter $D_1$ on $I$ which saturates ultrapowers of the random graph. "Saturates" means that $M^I/\de_1$ is $λ^+$-saturated whenever M is a model of the theory of the random graph. This was known to be true for stable theories, and false for non-simple and non-low theories. This result and the techniques introduced in the proof have catalyzed the authors' subsequent work on Keisler's order for simple unstable theories. The introduction, which includes a part written for model theorists and a part written for set theorists, discusses our current program and related results.

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arXiv2012

Model-theoretic properties of ultrafilters built by independent families of functions

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 have a precise degree of saturation. 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 the inductive construction of a regular ultrafilter on $λ$ (i.e. a point after which all antichains of $P(λ)/D$ have cardinality less than $λ$) essentially prevents any subsequent ultrafilter from being flexible, {thus} from saturating any non-low theory. The paper's three main constructions are as follows. First, we construct a regular filter $D$ on $λ$ so that any ultrafilter extending $D$ fails to $λ^+$-saturate ultrapowers of the random graph, {thus} of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal $κ$, we construct a regular ultrafilter on $λ> κ$ which is $λ$-flexible but not $κ^{++}$-good, improving our previous answer to a question raised in Dow 1975. Third, assuming a weakly compact cardinal $κ$, we construct an ultrafilter to show that $\lcf(\aleph_0)$ may be small while all symmetric cuts of cofinality $κ$ are realized. Thus certain families of pre-cuts may be realized while still failing to saturate any unstable theory.

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arXiv2012

A Dividing Line Within Simple Unstable Theories

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 $μ< λ\leq 2^μ$, we construct a regular ultrafilter D on $λ$ such that (i) for any model $M$ of a stable theory or of the random graph, $M^λ/D$ is $λ^+$-saturated but (ii) if $Th(N)$ is not simple or not low then $N^λ/D$ is not $λ^+$-saturated. The non-saturation result relies on the notion of flexible ultrafilters. To prove the saturation result we develop a property of a class of simple theories, called Qr1, generalizing the fact that whenever $B$ is a set of parameters in some sufficiently saturated model of the random graph, $|B| = λ$ and $μ< λ\leq 2^μ$, then there is a set $A$ with $|A| = μ$ such that any non-algebraic $p \in S(B)$ is finitely realized in $A$. In addition to giving information about simple unstable theories, our proof reframes the problem of saturation of ultrapowers in several key ways. We give a new characterization of good filters in terms of "excellence," a measure of the accuracy of the quotient Boolean algebra. We introduce and develop the notion of {moral} ultrafilters on Boolean algebras. We prove a so-called "separation of variables" result which shows how the problem of constructing ultrafilters to have a precise degree of saturation may be profitably separated into a more set-theoretic stage, building an excellent filter, followed by a more model-theoretic stage: building moral ultrafilters on the quotient Boolean algebra, a process which highlights the complexity of certain patterns, arising from first-order formulas, in certain Boolean algebras.

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arXiv2012

Constructing regular ultrafilters from a model-theoretic point of view

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 $\lcf(\aleph_0, \de)$ of $\aleph_0$ modulo $\de$, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, detected by non-low theories. Assuming $κ> \aleph_0$ is measurable, we construct a regular ultrafilter on $λ\geq 2^κ$ which is flexible (thus: ok) but not good, and which moreover has large $\lcf(\aleph_0)$ but does not even saturate models of the random graph. We prove that there is a loss of saturation in regular ultrapowers of unstable theories, and give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. Finally, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler's order, and by recent work of the authors on $SOP_2$. We prove that for any $n < ω$, assuming the existence of $n$ measurable cardinals below $λ$, there is a regular ultrafilter $D$ on $λ$ such that any $D$-ultrapower of a model of linear order will have $n$ alternations of cuts, as defined below. Moreover, $D$ will $λ^+$-saturate all stable theories but will not $(2^κ)^+$-saturate any unstable theory, where $κ$ is the smallest measurable cardinal used in the construction.

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