This expository article is based on two lectures given by the first author at the Fields Institute in the Fall 2021 Thematic Program on Trends in Pure and Applied Model Theory. We give a detailed proof of a qualitative version of the Mallaris-Shelah regularity lemma for stable graphs using only basic local stability theory and an ultraproduct construction. This proof strategy was first established by Malliaris and Pillay, and later simplified by Pillay. We provide some further simplifications, and also explain how the pseudofinite approach can be used to obtain a qualitative strengthening (compared to previous proofs) in terms of "functional error". To illustrate the extra leverage obtained by functional error, we give an elementary argument for extracting equipartitions from arbitrary partitions.

Abstract Previous work of the author and Wolf, and independently Chernikov and Towsner, showed that 3 3 $3$ -uniform hypergraphs of small slicewise VC-dimension admit homogeneous partitions. The results of Chernikov and Towsner did not produce explicit bounds, while the work of the author and Wolf relied on a strong version of hypergraph regularity, and consequently produced a Wowzer type bound on the size of the partition. This paper gives a new proof of this result, yielding epsilon ϵ $\epsilon$ -homogeneous partitions of size at most 2 Superscript 2 Super Superscript epsilon Super Super Superscript negative upper K 2 2 ϵ − K $2^{2^{\epsilon ^{-K}}}$ , where upper K K $K$ is a constant depending on the slicewise VC-dimension. This is the first in a series of papers about growth of regular partitions of 3 3 $3$ -uniform hypergraphs. Our main result is a crucial ingredient in the second and third papers in the series, which investigate the growth of two types of regular partitions in hereditary properties of 3 3 $3$ -uniform hypergraphs, namely the weak type first defined by Chung and Haviland–Thomason, and the stronger type first developed by Gowers, and independently, Frankl and Rödl.

The $k$-dimensional functional order property ($\text{FOP}_k$) is a combinatorial property of a $(k+1)$-partitioned formula. This notion arose in work of Terry and Wolf, which identified $\text{NFOP}_2$ as a ternary analogue of stability in the context of two finitary combinatorial problems related to hypergraph regularity and arithmetic regularity. In this paper we show $\text{NFOP}_k$ has equally strong implications in model-theoretic classification theory, where its behavior as a $(k+1)$-ary version of stability is in close analogy to the behavior of $k$-dependence as a $(k+1)$-ary version of $\text{NIP}$. Our results include several new characterizations of $\text{NFOP}_k$, including a characterization in terms of collapsing indiscernibles, combinatorial recharacterizations, and a characterization in terms of type-counting when $k=2$. As a corollary of our collapsing theorem, we show $\text{NFOP}_k$ is closed under Boolean combinations, and that $\text{FOP}_k$ can always be witnessed by a formula where all but one variable have length $1$. When $k=2$, we prove a composition lemma analogous to that of Chernikov and Hempel from the setting of $2$-dependence. Using this, we provide a new class of algebraic examples of $\text{NFOP}_2$ theories. Specifically, we show that if $T$ is the theory of an infinite dimensional vector space over a field $K$, equipped with a bilinear form satisfying certain properties, then $T$ is $\text{NFOP}_2$ if and only if $K$ is stable. Along the way we provide a corrected and reorganized proof of Granger's quantifier elimination and completeness results for these theories.

A regular partition $\mathcal{P}$ for a $3$-uniform hypergraph $H=(V,E)$ consists of a partition $V=V_1\cup \ldots \cup V_t$ and for each $ij\in {[t]\choose 2}$, a partition $K_2[V_i,V_j]=P_{ij}^1\cup \ldots \cup P_{ij}^{\ell}$, such that certain quasirandomness properties hold. The complexity of $\mathcal{P}$ is the pair $(t,\ell)$. In this paper we show that if a $3$-uniform hypergraph $H$ has $VC_2$-dimension at most $k$, then there is a regular partition $\mathcal{P}$ for $H$ of complexity $(t,\ell)$, where $\ell$ is bounded by a polynomial in the degree of regularity. This is a vast improvement on the bound arising from the proof of this regularity lemma in general, in which the bound generated for $\ell$ is of Wowzer type. This can be seen as a higher arity analogue of the efficient regularity lemmas for graphs and hypergraphs of bounded VC-dimension due to Alon-Fischer-Newman, Lovász-Szegedy, and Fox-Pach-Suk.

Given a finite relational language L, a hereditary L-property is a class H of finite L-structures closed under isomorphism and substructure. The speed of H is the function which sends an integer n≥1 to the number of distinct elements in H with underlying set {1,...,n}. In this paper we give a description of many new jumps in the possible speeds of a hereditary L-property, where L is any finite relational language. In particular, we characterize the jumps in the polynomial and factorial ranges, and show they are essentially the same as in the case of graphs. The results in the factorial range are new for all examples requiring a language of arity greater than two, including the setting of hereditary properties of k-uniform hypergraphs for k>2. Further, adapting an example of Balogh, Bollobás, and Weinreich, we show that for all k≥2, there are hereditary properties of k-uniform hypergraphs whose speeds oscillate between functions near the upper and lower bounds of the penultimate range, ruling out many natural functions as jumps in that range. Our theorems about the factorial range use model theoretic tools related to the notion of mutual algebraicity.

Suppose G is a finite group and A subset of G is such that {gA : g is an element of G} has VC-dimension strictly less than k. We find algebraically well-structured sets in epsilon which, up to a chosen epsilon > 0, describe the structure of A and behave regularly with respect to translates of A. For the subclass of groups with uniformly fixed finite exponent r, these algebraic objects are normal subgroups with index bounded in terms of k, r, and epsilon. For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model-theoretic, and heavily rely on a structural analysis of compactifications of pseudofinite groups as inverse limits of Lie groups. The introduction of Bohr neighborhoods into the nonabelian setting uses model theoretic methods related to the work of Breuillard, Green, and Tao [8] and Hrushovski [28] on approximate groups, as well as a result of Alekseev, Glebskii, and Gordon [1] on approximate homomorphisms.

Over the past several years, numerous authors have explored model theoretically motivated combinatorial conditions that ensure that a graph has an efficient regular decomposition in the sense of Szemer\'edi. In this paper we set out a research program that explores a corresponding set of questions for 3-uniform hypergraphs, a setting in which useful notions of regularity are significantly more intricate. The main results in this paper concern certain combinatorial properties which arose as natural higher-order generalizations of the order property in parallel work of the authors in the arithmetic setting. Interpreted in the context of 3-uniform hypergraphs, these are tightly connected to the nature of irregular triads. Specifically, we show that a hereditary property of 3-uniform hypergraphs admits regular decompositions with so-called"linear error"if and only if it does not have the functional order property. Along the way, we show that a hereditary property of 3-uniform hypergraphs is homogeneous (i.e. all regular triads have density near $0$ or near $1$) if and only it has bounded $\textrm{VC}_2$-dimension, complementing a recent (non-quantitative) result of Chernikov and Towsner. We also address several questions arising from prior work on tame regularity in hypergraphs. In particular, we characterize the hereditary properties of $3$-uniform hypergraphs admitting the type of regular partitions appearing in work of Fox et al. as those that have finite weak VC-dimension. This is again analogous to a recent non-quantitative result of Chernikov and Towsner.

. We investigate an open question concerning properties of algebraic independence in continuous theories (see Section 4). The rest of the work is essentially a translation to continuous logic of popular notions of independence (in particular, forking and dividing). Much of the time we are simply “copying” classical proofs from well-known sources, while along the way making the necessary adjustments for continuous languages.

We define a natural notion of higher-order stability and show that subsets of $\mathbb{F}_p^n$ that are tame in this sense can be approximately described by a union of low-complexity quadratic subvarieties up to linear error. This generalizes the arithmetic regularity lemma for stable subsets of $\mathbb{F}_p^n$ proved by the authors, as well as subsequent refinements and generalizations by the authors, and Conant, Terry, and Pillay, to the realm of higher-order Fourier analysis.

We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A qualitative version of this generalisation was recently obtained by the first author in joint work with Conant and Pillay, using model-theoretic techniques. In contrast, the approach in the present paper is highly quantitative and relies on several key ingredients from arithmetic combinatorics.

AbstractWe prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k-stable. Then there is a normal subgroup H ≤ G of index at most n, and a set Y ⊆ G, which is a union of cosets of H, such that |A △ Y| ≤ε|H|. It follows that, for any coset C of H, either |C ∩ A|≤ ε|H| or |C \ A| ≤ ε |H|. This qualitatively generalises recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$.

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if T is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.

An $(n,s,q)$-graph is an $n$-vertex multigraph where every set of $s$ vertices spans at most $q$ edges. In this paper, we determine the maximum product of the edge multiplicities in $(n,s,q)$-graphs if the congruence class of $q$ modulo ${s\choose 2}$ is in a certain interval of length about $3s/2$. The smallest case that falls outside this range is $(s,q)=(4,15)$, and here the answer is $a^{n^2+o(n^2)}$ where $a$ is transcendental assuming Schanuel's conjecture. This could indicate the difficulty of solving the problem in full generality. Many of our results can be seen as extending work by Bondy-Tuza and Furedi-Kundgen about sums of edge multiplicities to the product setting. We also prove a variety of other extremal results for $(n,s,q)$-graphs, including product-stability theorems. These results are of additional interest because they can be used to enumerate and to prove logical 0-1 laws for $(n,s,q)$-graphs. Our work therefore extends many classical enumerative results in extremal graph theory beginning with the Erdős-Kleitman-Rothschild theorem to multigraphs.

We prove that the number of multigraphs with vertex set {1, . . ., n } such that every four vertices span at most nine edges is a n 2 + o ( n 2 ) where a is transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov, who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy–Schwarz arguments. Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.

AbstractFix an integer $r \ge 3$. We consider metric spaces on n points such that the distance between any two points lies in $\left\{ {1, \ldots ,r} \right\}$. Our main result describes their approximate structure for large n. As a consequence, we show that the number of these metric spaces is $\left\lceil {{{r + 1} \over 2}} \right\rceil ^{\left( {\matrix{ n \cr 2 \cr } } \right) + o\left( {n^2 } \right)} .$Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij [34]. When r is even, our structural characterization is more precise and implies that almost all such metric spaces have all distances at least $r/2$. As an easy consequence, when r is even, we improve the error term above from $o\left( {n^2 } \right)$ to $o\left( 1 \right)$, and also show a labeled first-order 0-1 law in the language ${\cal L}_r $, consisting of r binary relations, one for each element of $[r]$ . In particular, we show the almost sure theory T is the theory of the Fraïssé limit of the class of all finite simple complete edge-colored graphs with edge colors in $\left\{ {r/2, \ldots ,r} \right\}$.Our work can be viewed as an extension of a long line of research in extremal combinatorics to the colored setting, as well as an addition to the collection of known structures that admit logical 0-1 laws.

The arithmetic regularity lemma for Fpn, proved by Green in 2005, states that given a subset A subset of Fpn, there exists a subspace H <= Fpn of bounded codimension such that A is Fourier-uniform with respect to almost all cosets of H. It is known that in general, the growth of the codimension of H is required to be of tower type depending on the degree of uniformity, and that one must allow for a small number of non-uniform cosets. Our main result is that, under a natural model-theoretic assumption of stability, the tower-type bound and non-uniform cosets in the arithmetic regularity lemma are not necessary. Specifically, we say that a set A subset of Fpn is k-stable if there are no a1, horizontal ellipsis ,ak,b1, horizontal ellipsis ,bk is an element of Fpn such that ai+bj is an element of A if and only if i <= j. We prove an arithmetic regularity lemma for k-stable subsets A subset of Fpn in which the bound on the codimension of the subspace is a polynomial (depending on k) in the degree of uniformity, and in which there are no non-uniform cosets. This result is an arithmetic analogue of the stable graph regularity lemma proved by Malliaris and Shelah.

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.

In this paper we investigate a connection between the growth rates of certain classes of finite structures and a generalization of V C \mathrm {VC} -dimension called V C ℓ \mathrm {VC}_{\ell } -dimension. Let L \mathcal {L} be a finite relational language with maximum arity r r . A hereditary L \mathcal {L} -property is a class of finite L \mathcal {L} -structures closed under isomorphism and substructures. The speed of a hereditary L \mathcal {L} -property H \mathcal {H} is the function which sends n n to | H n | |\mathcal {H}_n| , where H n \mathcal {H}_n is the set of elements of H \mathcal {H} with universe { 1 , … , n } \{1,\ldots , n\} . It was previously known that there exists a gap between the fastest possible speed of a hereditary L \mathcal {L} -property and all lower speeds, namely between the speeds 2 Θ ( n r ) 2^{\Theta (n^r)} and 2 o ( n r ) 2^{o(n^r)} . We strengthen this gap by showing that for any hereditary L \mathcal {L} -property H \mathcal {H} , either | H n | = 2 Θ ( n r ) |\mathcal {H}_n|=2^{\Theta (n^r)} or there is ϵ > 0 \epsilon >0 such that for all large enough n n , | H n | ≤ 2 n r − ϵ |\mathcal {H}_n|\leq 2^{n^{r-\epsilon }} . This improves what was previously known about this gap when r ≥ 3 r\geq 3 . Further, we show this gap can be characterized in terms of V C ℓ \mathrm {VC}_{\ell } -dimension, therefore drawing a connection between this finite counting problem and the model theoretic dividing line known as ℓ \ell -dependence.

Given a finite relational language $\calL$, a hereditary $\calL$-property is a class of finite $\calL$-structures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and results form the study of hereditary properties in combinatorics to the setting of hereditary $\calL$-properties, where $\calL$ is any finite relational language with maximum arity at least two. In particular, the goal of this paper is to generalize how extremal results and stability theorems can be combined with standard techniques and tools to yield approximate enumeration and structure theorems. We accomplish this by generalizing the notions of extremal graphs, asymptotic density, and graph stability theorems using structures in an auxiliary language associated to a hereditary $\calL$-property. Given a hereditary $\calL$-property $\calH$, we prove an approximate asymptotic enumeration theorem for $\calH$ in terms of its generalized asymptotic density. Further we prove an approximate structure theorem for $\calH$, under the assumption of that $\calH$ has a stability theorem. The tools we use include a new application of the hypergraph containers theorem (Balogh-Morris-Samotij, Saxton-Thomason) to the setting of $\calL$-structures, a general supersaturation theorem for hereditary $\calL$-properties (also new), and a general graph removal lemma for $\calL$-structures proved by Aroskar and Cummings.

In this paper we investigate a connection between the growth rates of certain classes of finite structures and a generalization of $\text{VC}$-dimension called $\text{VC}_{\ell}$-dimension. Let $\mathcal{L}$ be a finite relational language with maximum arity $r$. A hereditary $\mathcal{L}$-property is a class of finite $\mathcal{L}$-structures closed under isomorphism and substructures. The \emph{speed} of a hereditary $\mathcal{L}$-property $\mathcal{H}$ is the function which sends $n$ to $|\mathcal{H}_n|$, where $\mathcal{H}_n$ is the set of elements of $\mathcal{H}$ with universe $\{1,\ldots, n\}$. It was previously known there exists a gap between the fastest possible speed of a hereditary $\mathcal{L}$-property and all lower speeds, namely between the speeds $2^{\Theta(n^r)}$ and $2^{o(n^r)}$. We strengthen this gap by showing that for any hereditary $\mathcal{L}$-property $\mathcal{H}$, either $|\mathcal{H}_n|=2^{\Theta(n^r)}$ or there is $\epsilon>0$ such that for all large enough $n$, $|\mathcal{H}_n|\leq 2^{n^{r-\epsilon}}$. This improves what was previously known about this gap when $r\geq 3$. Further, we show this gap can be characterized in terms of $\text{VC}_{\ell}$-dimension, therefore drawing a connection between this finite counting problem and the model theoretic dividing line known as $\ell$-dependence.