Recent work of Gowers and Nagle, Rödl, Schacht, and Skokan has established a hypergraph removal lemma, which in turn implies some results of Szemerédi and Furstenberg-Katznelson concerning one-dimensional and multi-dimensional arithmetic progressions respectively. In this paper we shall give a self-contained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper to establish infinitely many constellations of a prescribed shape in the Gaussian primes.
We study the integral and measure theory of the ultraproduct of finite sets. As a main application we construct limit objects for hypergraph sequences. We give a new proof for the Hypergraph Removal Lemma and the Hypergraph Regularity Lemma.
We obtain a hypergraph generalisation of the graph blow-up lemma proved by Komlos, Sarkozy and Szemeredi, showing that hypergraphs with sufficient regularity and no atypical vertices behave as if they were complete for the purpose of embedding bounded degree hypergraphs.
We give an extension of a graph result by Alon and Shapira. And it affirmatively settles a question on property testing raised by them. All monotone hypergraph properties and all hereditary partite hypergraph properties are testable. Our proof is constructive and based on a new hypergraph regularity lemma.
Green [Geometric and Functional Analysis 15 (2005), 340--376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.
In this paper we develop a measure-theoretic method to treat problems in hypergraph theory. Our central theorem is a correspondence principle between three objects: An increasing hypergraph sequence, a measurable set in an ultraproduct space and a measurable set in a finite dimensional Lebesgue space. Using this correspondence principle we build up the theory of dense hypergraphs from scratch. Along these lines we give new proofs for the Hypergraph Removal Lemma, the Hypergraph Regularity Lemma, the Counting Lemma and the Testability of Hereditary Hypergraph Properties. We prove various new re…
We present a general framework to represent discrete configuration systems using hypergraphs. This representation allows one to transfer combinatorial removal lemmas to their analogues for configuration systems. These removal lemmas claim that a system without many configurations can be made configuration-free by removing a few of its constituent elements. As applications of this approach we give, an alternative proof of the removal lemma for permutations by Cooper, a general version of a removal lemma for linear systems in finite abelian groups, an interpretation of the mentioned removal lemm…
The classical sharp threshold theorem of Friedgut and Kalai (1996) asserts that any symmetric monotone function $f:\{0,1\}^{n}\to\{0,1\}$ exhibits a sharp threshold phenomenon. This means that the expectation of $f$ with respect to the biased measure $μ_{p}$ increases rapidly from 0 to 1 as $p$ increases. In this paper we present `robust' versions of the theorem, which assert that it holds also if the function is `almost' monotone, and admits a much weaker notion of symmetry. Unlike the original proof of the theorem which relies on hypercontractivity, our proof relies on a `regularity' lemma (…
The hypergraph regularity lemma -- the extension of Szemerédi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-Rödl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $k \ge 2$, thus confirming a prediction of Tao. Pri…
For a graph $F$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ (or a Berge-$F$ in short), if there is a bijection $f : E(F) \rightarrow E(\mathcal{H})$ such that for each $e \in E(F)$ we have $e \subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain a Berge copy of $F$. We denote the maximum number of hyperedges in an $n$-vertex $r$-uniform Berge-$F$-free hypergraph by $\mathrm{ex}_r(n,\textrm{Berge-}F).$ In this paper we prove two general lemmas concerning the maximum size of a Berge-$F$-free hypergraph and use them to establish new results and improve several old results. I…
The hypergraph regularity lemma -- the extension of Szemerédi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-Rödl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. In a recent paper we have shown that these bounds are unavoidable for $3$-uniform hypergraphs. In this paper we exten…
A common theme in many extremal problems in graph theory is the relation between local and global properties of graphs. One of the most celebrated results of this type is the Ruzsa-Szemerédi triangle removal lemma, which states that if a graph is $\varepsilon$-far from being triangle free, then most subsets of vertices of size $C(\varepsilon)$ are not triangle free. Unfortunately, the best known upper bound on $C(\varepsilon)$ is given by a tower-type function, and it is known that $C(\varepsilon)$ is not polynomial in $\varepsilon^{-1}$. The triangle removal lemma has been extended to many ot…
We prove analogues for hypergraphs of Szemerédi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemerédi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound. Similar results with the same consequences have been obtained independently by Nagle, Rödl, Schacht and Skokan.
We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon--Ben-Eliezer--Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph $(V,G,<)$ has few induced copies of a small ordered hypergraph $(W,H,\prec)$ then there is a small modification $G'$ so that $(V,G',<)$ has no induced copies of $(W,H,\prec)$. (Note that we do \emph{not} need to modify the ordering $<$.) We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstr…
We prove a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities. The main novelty is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry. The existence of smaller families of containers for independent sets in such hypergraphs, which is guaranteed by the new theorem, allows us to improve upon the best currently known bounds for several problems in extremal graph theory, discrete geometry, and Ramsey theory.
As an application of Szemeredi's regularity lemma, Erdos-Frankl-Rodl (1986) showed that the number of graphs on vertex set {1,2,...n} with a monotone class P is $2^{(1+o(1))ex(n,P)n^2/2}$ where $ex(n,P)$ is the maximum number of edges of an n-vertex graph which has no subgraph in P. Kohayakawa et al. (2003) extended it from monotone to hereditary and from graphs to 3-uniform hypergraphs. We extend it to general hypergraphs. This may be a simple example illustrating how to apply a recent hypergraph regularity lemma by the author.
The clique removal lemma says that for every $r \geq 3$ and $\varepsilon>0$, there exists some $δ>0$ so that every $n$-vertex graph $G$ with fewer than $δn^r$ copies of $K_r$ can be made $K_r$-free by removing at most $\varepsilon n^2$ edges. The dependence of $δ$ on $\varepsilon$ in this result is notoriously difficult to determine: it is known that $δ^{-1}$ must be at least super-polynomial in $\varepsilon^{-1}$, and that it is at most of tower type in $\log \varepsilon^{-1}$. We prove that if one imposes an appropriate minimum degree condition on $G$, then one can actually take $δ$ to be a …
The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph $H$ and $\varepsilon > 0$, if an $n$-vertex graph $G$ contains $\varepsilon n^2$ edge-disjoint copies of $H$ then $G$ contains $δn^{v(H)}$ copies of $H$ for some $δ= δ(\varepsilon,H) > 0$. The current proofs of the removal lemma give only very weak bounds on $δ(\varepsilon,H)$, and it is also known that $δ(\varepsilon,H)$ is not polynomial in $\varepsilon$ unless $H$ is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that $δ(\vareps…
The use of tools from analysis to approach problems in graph theory has become an active area of research. Usually such methods are applied to problems involving dense graphs and hypergraphs; here we give the an extension of such methods to sparse but pseudorandom hypergraphs. We use this framework to give a proof of hypergraph removal for sub-hypergraphs of sparse random hypergraphs.
We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $ε>0$ there exists $M$ such that every triangle-free graph $G$ has an $ε$-approximate homomorphism to a triangle-free graph $F$ on at most $M$ vertices (here an $ε$-approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $ε|V(G)|^2$ edges of $G$ are mapped to edges of $F$). One consequence of our results is that the least possible $M$ in the triangle-free lemma grows faster than exponential in any polyn…
Let $G$ be an abelian group of bounded exponent and $A \subseteq G$. We show that if the collection of translates of $A$ has VC dimension at most $d$, then for every $ε>0$ there is a subgroup $H$ of $G$ of index at most $ε^{-d-o(1)}$ such that one can add or delete at most $ε|G|$ elements to/from $A$ to make it a union of $H$-cosets. We also establish a removal lemma with polynomial bounds, with applications to property testing, for induced bipartite patterns in a finite abelian group with bounded exponent.
Obtaining an efficient bound for the triangle removal lemma is one of the most outstanding open problems of extremal combinatorics. Perhaps the main bottleneck for achieving this goal is that triangle-free graphs can be highly unstructured. For example, triangle-free graphs might have only regular partitions (in the sense of Szemerédi) of tower-type size. And indeed, essentially all the graph properties ${\cal P}$ for which removal lemmas with reasonable bounds were obtained, are such that every graph satisfying ${\cal P}$ has a small regular partition. So in some sense, a barrier for obtainin…
Covering arrays are generalizations of orthogonal arrays that have been widely studied and are used in software testing. The probabilistic method has been employed to derive upper bounds on the sizes of minimum covering arrays and give asymptotic upper bounds that are logarithmic on the number of columns of the array. This corresponds to test suites with a desired level of coverage of the parameter space where we guarantee the number of test cases is logarithmic on the number of parameters of the system. In this paper, we study variable strength covering arrays, a generalization of covering ar…
We present a systematic study of the regularity phenomena for NIP hypergraphs and connections to the theory of (locally) generically stable measures, providing a model-theoretic hypergraph version of the results from [L. Lovász, B. Szegedy, "Regularity partitions and the topology of graphons", An irregular mind, Springer Berlin Heidelberg, 2010, 415-446]. Besides, we revise the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the results from [A. Chernikov, S. Starchenko, "Regularity lemma for distal structures", J. Eur. Math. Soc. 20 (2018), 2437-…
The triangle removal lemma states that a simple graph with o(n^3) triangles can be made triangle-free by removing o(n^2) edges. It is natural to ask if this widely used result can be extended to multi-graphs (or equivalently, weighted graphs). In this short paper we rule out the possibility of such an extension by showing that there are multi-graphs with only n^{2+o(1)} triangles that are still far from being triangle-free. On the other hand, we show that for some g(n)=ω(1), if a multi-graph (or weighted graph) has only g(n)n^2 triangles then it must be close to being triangle-free. The proof …
Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for semi-algebraic $k$-uniform hypergraphs of bounded complexity, showing that for each $ε>0$ the vertex set can be equitably partitioned into a bounded number of parts (in terms of $ε$ and the complexity) so that all but an $ε$-fraction of the $k$-tuples of parts are homogeneous. We prove that the number of parts can be taken to be polynomial in $1/ε$. Our improved regularity lemma can be applied to geometric problems and to the following general question on property testing: is it possible to decide, with query complexity polyn…
A recent result of Alon, Ben-Eliezer and Fischer establishes an induced removal lemma for ordered graphs. That is, if $F$ is an ordered graph and $\varepsilon>0$, then there exists $δ_{F}(\varepsilon)>0$ such that every $n$-vertex ordered graph $G$ containing at most $δ_{F}(\varepsilon) n^{v(F)}$ induced copies of $F$ can be made induced $F$-free by adding/deleting at most $\varepsilon n^2$ edges. We prove that $δ_{F}(\varepsilon)$ can be chosen to be a polynomial function of $\varepsilon$ if and only if $|V(F)|=2$, or $F$ is the ordered graph with vertices $x<y<z$ and edges $\{x,y\},\{x,z\}$ …
The Lovász Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for colouring graphs and hypergraphs with bounded maximum degree. This paper presents a general theorem for colouring hypergraphs that in many instances matches or slightly improves upon the bounds obtained using the Lovász Local Lemma. Moreover, the theorem directly shows that there are exponentially many colourings. The elementary and self-contained proof is inspired by a recent result for nonrepetitive colourings by Rosenfeld [2020]. We apply our general t…
Szemerédi's regularity lemma is a powerful tool in graph theory. It states that for every large enough graph, there exists a partition of the edge set with bounded size such that most induced subgraphs are quasirandom. When the graph is a definable set $ϕ(x, y)$ in a finite field $F_q$, Tao's algebraic graph regularity lemma shows that there is a partition of the graph $ϕ(x, y)$ such that all induced subgraphs are quasirandom and the error bound on quasirandomness is $O(q^{-1/4})$. In this work we prove an algebraic hypergraph regularity lemma for definable sets in finite fields, thus answerin…
We prove an arithmetic removal result for all compact abelian groups, generalizing a finitary removal result of Král', Serra and the third author. To this end, we consider infinite measurable hypergraphs that are invariant under certain group actions, and for these hypergraphs we prove a symmetry-preserving removal lemma, which extends a finitary result of the same name by the second author. We deduce our arithmetic removal result by applying this lemma to a specific type of invariant measurable hypergraph. As a direct application, we obtain the following generalization of Szemerédi's theorem:…
We obtain a removal lemma for systems of linear equations over the circle group, using a similar result for finite fields due to Král, Serra and Vena, and we discuss some applications.
Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(n^h) copies of H can be made H-free by removing o(n^2) edges. We give a new proof which avoids Szemerédi's regularity lemma and gives a better bound. This approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma. This answers questions of Alon and Gowers.
We formulate and prove inverse mixing lemmas in the settings of simplicial complexes and k-uniform hypergraphs. In the hypergraph setting, we extend results of Bilu and Linial for graphs. In the simplicial complex setting, our results answer a question of Parzanchevski et al.
The graph removal lemma states that any graph on n vertices with o(n^{v(H)}) copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.
We introduce a correspondence principle (analogous to the Furstenberg correspondence principle) that allows one to extract an infinite random graph or hypergraph from a sequence of increasingly large deterministic graphs or hypergraphs. As an application we present a new (infinitary) proof of the hypergraph removal lemma of Nagle-Schacht-Rödl-Skokan and Gowers, which does not require the hypergraph regularity lemma and requires significantly less computation. This in turn gives new proofs of several corollaries of the hypergraph removal lemma, such as Szemerédi's theorem on arithmetic progress…
In this paper, we develop a method for studying cycle lengths in hypergraphs. Our method is built on earlier ones used in [21,22,18]. However, instead of utilizing the well-known lemma of Bondy and Simonovits [4] that most existing methods do, we develop a new and very simple lemma in its place. One useful feature of the new lemma is its adaptiveness for the hypergraph setting. Using this new method, we prove a conjecture of Verstraëte [37] that for $r\ge 3$, every $r$-uniform hypergraph with average degree $Ω(k^{r-1})$ contains Berge cycles of $k$ consecutive lengths. This is sharp up to the …
We show that the Ramsey number is linear for every uniform hypergraph with bounded-degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. showed in 1983. Our proof is simple, contains the multicolor case, and provides a strong embedding lemma. It shows the potential of a new hypergraph regularity lemma by the author.
Homogeneous hypergraph regularity lemmas via $k$-strong honest definitions
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom（剑桥大学数学科学中心纯数学与数理统计系）
Suppose one needs to change the direction of at least $εn^2$ edges of an $n$-vertex tournament $T$, in order to make it $H$-free. A standard application of the regularity method shows that in this case $T$ contains at least $f^*_H(ε)n^h$ copies of $H$, where $f^*_H$ is some tower-type function. It has long been observed that many graph/digraph problems become easier when assuming that the host graph is a tournament. It is thus natural to ask if the removal lemma becomes easier if we assume that the digraph $G$ is a tournament. Our main result here is a precise characterization of the tournamen…
We present a variant of a universality result of Rödl [On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1986), no. 1-2, 125-134] for sparse, $3$-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ``pseudorandom'' uniform hypergraphs, which is proved in the companion paper [Counting results for sparse pseudorandom hypergraphs I].
In this note we observe that in the hyper-graph removal lemma the edge removal can be done in a way that the symmetries of the original hyper-graph remain preserved. As an application we prove the following generalization of Szemerédi's Theorem on arithmetic progressions. If in an Abelian group $A$ there are sets $S_1,S_2...,S_t$ such that the number of arithmetic progressions $x_1,x_2,...,x_t$ with $x_i\in S_i$ is $o(|A|^2)$ then we can shrink each $S_i$ by $o(|A|)$ elements such that the new sets don't have such a diagonal arithmetic progression.
We introduce a new variant of Szemerédi's regularity lemma which we call the "sparse regular approximation lemma" (SRAL). The input to this lemma is a graph $G$ of edge density $p$ and parameters $ε, δ$, where we think of $δ$ as a constant. The goal is to construct an $ε$-regular partition of $G$ while having the freedom to add/remove up to $δ|E(G)|$ edges. As we show here, this weaker variant of the regularity lemma already suffices for proving the graph removal lemma and the hypergraph regularity lemma, which are two of the main applications of the (standard) regularity lemma. This of course…
We study sparse hypergraphs which satisfy a mild pseudorandomness condition known as $L_p$ regularity. We prove appropriate regularity and counting lemmas, and we extend the relative removal lemma of Tao in this setting. This answers a question of Borgs, Chayes, Cohn and Zhao.
We give a fully polynomial-time approximation scheme (FPTAS) to count the number of $q$-colorings for $k$-uniform hypergraphs with maximum degree $Δ$ if $k\ge 28$ and $q >357Δ^{\frac{14}{k-14}}$ . We also obtain a polynomial-time almost uniform sampler if $q>931Δ^{\frac{16}{k-16/3}}$. These are the first approximate counting and sampling algorithms in the regime $q\llΔ$ (for large $Δ$ and $k$) without any additional assumptions. Our method is based on the recent work of Moitra (STOC, 2017). One important contribution of ours is to remove the dependency of $k$ and $Δ$ in Moitra's approach.
Let $p$ be a fixed prime. A triangle in $\mathbb{F}_p^n$ is an ordered triple $(x,y,z)$ of points satisfying $x+y+z=0$. Let $N=p^n=|\mathbb{F}_p^n|$. Green proved an arithmetic triangle removal lemma which says that for every $ε>0$ and prime $p$, there is a $δ>0$ such that if $X,Y,Z \subset \mathbb{F}_p^n$ and the number of triangles in $X \times Y \times Z$ is at most $δN^2$, then we can delete $εN$ elements from $X$, $Y$, and $Z$ and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a…
We prove a removal lemma for systems of linear equations over finite fields: let $X_1,...,X_m$ be subsets of the finite field $\F_q$ and let $A$ be a $(k\times m)$ matrix with coefficients in $\F_q$ and rank $k$; if the linear system $Ax=b$ has $o(q^{m-k})$ solutions with $x_i\in X_i$, then we can destroy all these solutions by deleting $o(q)$ elements from each $X_i$. This extends a result of Green [Geometric and Functional Analysis 15(2) (2005), 340--376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for syste…
This is a companion note to the paper "Almost all Steiner triple systems have perfect matchings (arXiv:1611.02246). That paper contains several general lemmas about random Steiner triple systems; in this note we record analogues of these lemmas for random Latin squares, which in particular are necessary ingredients for our recent paper "Large deviations in random Latin squares" (arXiv:2106.11932). Most important is a relationship between uniformly random order-$n$ Latin squares and the triangle removal process on the complete tripartite graph $K_{n,n,n}$.
Our work builds on known results for k-uniform hypergraphs including the existence of limits, a Regularity Lemma and a Removal Lemma. Our main tool here is a theory of measures on ultraproduct spaces which establishes a correspondence between ultraproduct spaces and Euclidean spaces. First we show the existence of a limit object for convergent sequences of relational structures and as a special case, we retrieve the known limits for graphs and digraphs. Then we extend this notion to finite models of a fixed universal theory. We also state and prove a Regularity Lemma and a Removal Lemma. We wi…
In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the $k$--determinantal of an integer $(k\times m)$ matrix $A$ is coprime with the order $n$ of a group $G$ and the number of solutions of the system $Ax=b$ with $x_1\in X_1,..., x_m\in X_m$ is $o(n^{m-k})$, then we can eliminate $o(n)$ elements in each set to remove all these solutions. This is a follow-up of our former paper 'A Removal Lemma for Systems of Linear Equations over Finite Fields' arXiv:0809.1846v1, which dealt with the case of finite fields.
We complete the characterization of the digraphs $D$ for which the induced $D$-removal lemma has polynomial bounds, answering a question of Alon and Shapira. We also study the analogous problem for $k$-colored complete graphs. In particular, we prove a removal lemma with polynomial bounds for Gallai colorings.
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 ε-regular, where c,ε>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 Szemerédi'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, w…
For each $k\geq 3$, Green proved an arithmetic $k$-cycle removal lemma for any abelian group $G$. The best known bounds relating the parameters in the lemma for general $G$ are of tower-type. For $k>3$, even in the case $G=\mathbb{F}_2^n$ no better bounds were known prior to this paper. This special case has received considerable attention due to its close connection to property testing of boolean functions. For every $k\geq 3$, we prove a polynomial bound relating the parameters for $G=\mathbb{F}_p^n$, where $p$ is any fixed prime. This extends the result for $k=3$ by the first two authors. D…
