M. Malliaris

General topology meets model theory, on p and t.

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 25(4):220–224] and Cohen [Cohen P (1963) Proc Natl Acad Sci USA 50(6):1143–1148], Hilbert’s first problem is independent of ZFC (Zermelo-Fraenkel set theory with the axiom of choice). Much work both before and since has been done on inequalities between these cardinal invariants, but some basic questions have remained open despite Cohen’s introduction of forcing. The oldest and perhaps most famous of these is whether “,” which was proved in a special case by Rothberger [Rothberger F (1948) Fund Math 35:29–46], building on Hausdorff [Hausdorff (1936) Fund Math 26:241–255]. In this paper we explain how our work on the structure of Keisler’s order, a large-scale classification problem in model theory, led to the solution of this problem in ZFC as well as of an a priori unrelated open question in model theory.

CONSTRUCTING REGULAR ULTRAFILTERS FROM A MODEL-THEORETIC POINT OF VIEW

This paper contributes to the set-theoretic side of understanding Keislers order. We consider properties of ultralters which aect saturation of unstable theories: the lower conality lcf(@0,D) of @0 modulo D, saturation of the minimum unstable theory (the random graph), exi- bility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultralters thus assume a measurable car- dinal. The main results are as follows. First, we investigate the strength of exibility, known to be detected by non-low theories. Assuming� > @0 is measurable, we construct a regular ultralter on � � 2 � which is exible but not good, and which moreover has large lcf(@0) but does not even saturate models of the random graph. This implies (a) that exibility alone cannot characterize saturation of any theory, however (b) by separating exibility from goodness, we remove a main obstacle to proving non-low does not imply maximal. Since exible is precisely OK, this also shows that (c) from a set-theoretic point of view, consistently, ok need not imply good, addressing a prob- lem from Dow 1985. Second, under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and also give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. More precisely, for D regular onand M a model of an unstable theory, M � /D is not (2 � ) + -saturated; and for M a model of a non-simple theory and � = � <� , M � /D is not � ++ -saturated. In the third part of the paper, we investigate realization and omission of symmetric cuts, signicant both because of the maximality of the strict order property in Keislers order, and by recent work of the authors on SOP2. We prove that if D is a �-complete ultralter on �, any ultrapower of a suciently saturated model of linear or der will have no (�,�)-cuts, and that ifD is also normal, it will have a (� + ,� + )-cut. We apply this to prove that for any n < !, assuming the existence of n measurable cardinals below �, there is a regular ultralter D onsuch that any D-ultrapower of a model of linear order will haven alternations of cuts, as dened below. Moreover, D will � + -saturate all stable theories but will not (2 � ) + -saturate any unstable theory, whereis the smallest measurable cardinal used in the construction.

Independence, order, and the interaction of ultrafilters and theories

Abstract We consider the question, of longstanding interest, of realizing types in regular ultrapowers. In particular, this is a question about the interaction of ultrafilters and theories, which is both coarse and subtle. By our prior work it suffices to consider types given by instances of a single formula. In this article, we analyze a class of formulas φ whose associated characteristic sequence of hypergraphs can be seen as describing realization of first- and second-order types in ultrapowers on one hand, and properties of the corresponding ultrafilters on the other. These formulas act, via the characteristic sequence, as points of contact with the ultrafilter D , in the sense that they translate structural properties of ultrafilters into model-theoretically meaningful properties and vice versa. Such formulas characterize saturation for various key theories (e.g. T r g , T f e q ), yet their scope in Keisler’s order does not extend beyond T f e q . The proof applies Shelah’s classification of second-order quantifiers.

Edge distribution and density in the characteristic sequence

Abstract The characteristic sequence of hypergraphs 〈 P n : n ω 〉 associated to a formula φ ( x ; y ) , introduced in Malliaris (2010) [5] , is defined by P n ( y 1 , … , y n ) = ( ∃ x ) ⋀ i ≤ n φ ( x ; y i ) . We continue the study of characteristic sequences, showing that graph-theoretic techniques, notably Szemeredi’s celebrated regularity lemma, can be naturally applied to the study of model-theoretic complexity via the characteristic sequence. Specifically, we relate classification-theoretic properties of φ and of the P n (considered as formulas) to density between components in Szemeredi-regular decompositions of graphs in the characteristic sequence. In addition, we use Szemeredi regularity to calibrate model-theoretic notions of independence by describing the depth of independence of a constellation of sets and showing that certain failures of depth imply Shelah’s strong order property S O P 3 ; this sheds light on the interplay of independence and order in unstable theories.

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.

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.

What Simplicity Is Not

There is a duality in mathematics between proofs and counterexamples. To understand a mathematical question one investigates the limits. To investigate Hilbert’s 24th problem, and a mathematical concept of simplicity of a proof we deal here with both sides, focusing on what simplicity is not.