Classification theory and the number of non-isomorphic models

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck
 2/log*
 k pairs of parts which are not $${\epsilon}$$
 -regular, where $${c,\epsilon >0 }$$
 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemeredi’s regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter $${\epsilon}$$
 may require as many as $${2^{\Omega}(\epsilon^{-2})}$$
 parts. This is tight up to the implied constant and solves a problem studied by Lovasz and Szegedy.

/pdf/bounds-for-graph-regularity-and-removal-lemmas-276sepfu9v.pdf

Bounds for graph regularity and removal lemmas

We develop a framework in which Szemeredi'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 stability theory apply. In one direction, we address a problem from the classical Szemeredi theory. It was known that the "irregular pairs" in the statement of Szemeredi's regularity lemma cannot be eliminated, due to the counterexample of half-graphs (i.e., the order property, corresponding to model-theoretic instability). We show that half-graphs are the only essential difficulty, by giving a much stronger version of Szemeredi's regularity lemma for models of stable theories of graphs (i.e. graphs with the non-k�-order property), in which there are no irregular pairs, the bounds are significantly improved, and each component satisfies an indivisibility condition. In the other direction, we take a more model-theoretic approach, and give several new Szemeredi-type partition theorems for models of stable theories of graphs. The first theorem gives a partition of any such graph into indiscernible components, meaning here that each component is either a complete or an empty graph, whose interaction is strongly uniform. This relies on a finitary version of the classic model-theoretic fact that stable theories admit large sets of indiscernibles, by showing that in models of stable theories of graphs one can extract much larger indiscernible sets than expected by Ramsey's theorem. The second and third theorems allow for a much smaller number of components at the cost of weakening the "indivisibility" condition on the components. We also discuss some extensions to graphs without the independence property. All graphs are finite and all partitions are equitable, i.e. the sizes of the components differ by at most 1. In the last three theorems, the number of components depends on the size of the graph; in the first theorem quoted, this number is a function ofonly as in the usual Szemeredi regularity lemma.

/pdf/regularity-lemmas-for-stable-graphs-39n22km6c4.pdf

Regularity lemmas for stable graphs

https://escholarship.org/content/qt4h93b5rh/qt4h93b5rh.pdf?t=o0h7nr

Theories without the tree property of the second kind

We initiate a systematic study of the class of theories without the tree property of the second kind - NTP2. Most importantly, we show: the burden is "sub-multiplicative" in arbitrary theories (in particular, if a theory has TP2 then there is a formula with a single variable witnessing this); NTP2 is equivalent to the generalized Kim's lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters - so the dp-rank of a 1-type in any theory is always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic (0,0) is NTP2 (strong, of finite burden) if and only if the residue field is NTP2 (the residue field and the value group are strong, of finite burden respectively), so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2.

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 the class of thresholds over ℕ can not be learned in a private manner; this resolves open questions due to [Bun et al. 2015] and [Feldman and Xiao, 2015]. We leave as an open question whether every class with a finite Littlestone dimension can be learned by an approximately differentially private algorithm.

Private PAC learning implies finite Littlestone dimension

We connect and solve two longstanding open problems in quite different areas: the modeltheoretic question of whether SOP2 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 fundamental problem which we state and solve completely, using model-theoretic methods. By a cofinality spectrum problem s we essentially mean the data of a pair of models M M1 which code sufficient set theory, possibly in an expanded language, along with a distinguished set of formulas ∆s which define linear orders in M1. Let ts, the “treetops” of s, be the smallest regular cardinal λ such that one of a set of derived trees in M1 has a strictly increasing λ-sequence with no upper bound. Let C(s, ts) be the set of pairs of regular cardinals (κ1, κ2) such that κ1 ≤ κ2 λ,” or what is equivalent, such that (ω,<)λ/D contains no (κ, κ)-cuts for κ = cf(κ) ≤ λ, is λ+-good. We obtain several consequences, notably existence of a minimum Keisler class among the non-simple theories.

/pdf/cofinality-spectrum-theorems-in-model-theory-set-theory-and-vyjrvp6icb.pdf

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

Let T₁, T₂ be countable first-order theories, Mi ⊨ Ti, and &#119967; any regular ultrafilter on λ ≥ ℵ₀. A longstanding open problem of Keisler asks when T₂ is more complex than T₁, as measured by the fact that for any such λ, &#119967;, if the ultrapower (M₂)λ/&#119967; realizes all types over sets of size ≤ λ, then so must the ultrapower (M₁)λ/&#119967;. In this paper, building on the author's prior work [12] [13] [14], we show that the relative complexity of first-order theories in Keisler's sense is reflected in the relative graph-theoretic complexity of sequences of hypergraphs associated to formulas of the theory. After reviewing prior work on Keisler's order, we present the new construction in the context of ultrapowers, give various applications to the open question of the unstable classification, and investigate the interaction between theories and regularizing sets. We show that there is a minimum unstable theory, a minimum TP₂ theory, and that maximality is implied by the density of certain graph edges (between components arising from Szemeredi-regular decompositions) remaining bounded away from 0,1. We also introduce and discuss flexible ultrafilters, a relevant class of regular ultrafilters which reflect the sensitivity of certain unstable (non low) theories to the sizes of regularizing sets, and prove that any ultrafilter which saturates the minimal TP₂ theory is flexible.

/pdf/hypergraph-sequences-as-a-tool-for-saturation-of-ultrapowers-6q8kwo4zwk.pdf

Maryanthe Malliaris

Papers

Regularity lemmas for stable graphs

Private PAC learning implies finite Littlestone dimension

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

Hypergraph sequences as a tool for saturation of ultrapowers

Existence of optimal ultrafilters and the fundamental complexity of simple theories