A combinatorial proof of the Removal Lemma for Groups

The Minimum Degree Removal Lemma Thresholds

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…

Removal lemmas and approximate homomorphisms

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…

Efficient arithmetic regularity and removal lemmas for induced bipartite patterns

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.

General removal lemma

We formulate and prove a general result in spirit of hypergraph removal lemma for measurable functions of several variables.

Efficient Removal without Efficient Regularity

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…

有序匹配的多项式移除引理

Polynomial removal lemma for ordered matchings

本文证明了有序匹配的多项式移除引理，解决了Tomon与第一作者猜想的特例，并将结论推广到一致超图，即ε远不含H的有序图含poly(ε)n^t个H拷贝。

Multigraphs (only) satisfy a weak triangle removal lemma

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 …

Polynomial removal lemmas for ordered graphs

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 $\{x,y\},\{x,z\}$ …

A removal lemma for linear configurations in subsets of the circle

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.

A new proof of the graph removal lemma

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.

Graph removal lemmas

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.

Hypergraph Removal Lemmas via Robust Sharp Threshold Theorems

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 (…

广义组合Lason-Alon-Zippel-Schwartz零点定理引理

The Generalized Combinatorial Lason-Alon-Zippel-Schwartz Nullstellensatz Lemma

本文综述了Alon组合零点定理与Schwartz-Zippel引理的若干强化与推广，证明了当变量在足够大范围内独立取值时多元多项式非零值的存在性。

On a generalization of Kelly's combinatorial lemma

Kelly's combinatorial lemma is a basic tool in the study of Ulam's reconstruction conjecture. A generalization in terms of a family of t-elements subsets of a v-element set was given by Pouzet. We consider a version of this generalization modulo a prime p. We give illustrations to graphs and tournaments.

具有多项式界的超图移除

Hypergraph removal with polynomial bounds

本文证明了Kohayakawa、Nagle和Rödl的猜想：对任意$k>2$，只有$k$-部超图$F$才具有多项式界的超图移除引理，从而将Alon关于图的结论推广到一般超图。

The Removal Lemma for Tournaments

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…

A Removal Lemma for Ordered Hypergraphs

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…

The Symmetry Preserving Removal Lemma

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.

A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma

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…

A Removal Lemma for Systems of Linear Equations over Finite Fields

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…

三角形移除引理的一个改进构造

An improved construction for the triangle removal lemma

本文通过加性组合学中的角点问题思想，改进了三角形移除引理的构造，将指数常数从原有值提升至约1.6601，但未为角点问题本身带来新界。

A tight bound for Green's arithmetic triangle removal lemma in vector spaces

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…

Note on random Latin squares and the triangle removal process

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 $K_{n,n,n}$.

Limits, Regularity and Removal for Finite Structures

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…

通过双星形移动证明 Fan 引理

Fan's lemma via bistellar moves

本文将 Pachner 的双星形移动定理推广到带自由 Z2 作用的组合流形，并据此给出 Fan 引理的组合证明，且在三维及以下无需额外三角剖分假设。

On the Removal Lemma for Linear Systems over Abelian Groups

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.

Bounds for graph regularity and removal lemmas

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…

A Geometric Approach to Combinatorial Fixed-Point Theorems

We develop a geometric framework that unifies several different combinatorial fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing them to be different geometric manifestations of the same topological phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like fixed-point theorems involving an exponential-sized label set; (2) a generalization of Fan's parity proof of Tucker's Lemma to a much broader class of label sets; and (3) direct proofs of several Sperner-like lemmas from Tucker's lemma via explicit geometric embeddings, without the need for topological f…

A Characterization of Easily Testable Induced Digraphs and $k$-Colored Graphs

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.

A polynomial bound for the arithmetic $k$-cycle removal lemma in vector spaces

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…

Removal and Stability for Erdős-Ko-Rado

A $k$-uniform family of subsets of $[n]$ is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erdős-Ko-Rado theorem of 1961 that bounds the size of the largest such families. A recent trend has been to investigate the structure of set families with few disjoint pairs. Friedgut and Regev proved a general removal lemma, showing that when $γn \le k \le (\tfrac12 - γ)n$, a set family with few disjoint pairs can be made intersecting by removing few sets. We provide a simple proof of a removal…

A Combinatorial Analog of a Theorem of F.J.Dyson

Tucker's Lemma is a combinatorial analog of the Borsuk-Ulam theorem and the case n=2 was proposed by Tucker in 1945. Numerous generalizations and applications of the Lemma have appeared since then. In 2006 Meunier proved the Lemma in its full generality in his Ph.D. thesis. There are generalizations and extensions of the Borsuk-Ulam theorem that do not yet have combinatorial analogs. In this note, we give a combinatorial analog of a result of Freeman J. Dyson and show that our result is equivalent to Dyson's theorem. As with Tucker's Lemma, we hope that this will lead to generalizations and ap…

丰度：非对称图移除引理与线性方程的整数解

Abundance: Asymmetric Graph Removal Lemmas and Integer Solutions to Linear Equations

本文证明一大类图对满足非对称图移除引理的多项式依赖，构造出任意色数至少为4的K_t-丰度图，并将亏格至少为二的方程无互异整数解集合大小改进到最优的O(√N)。

广义Z_p-图克勒引理的组合度版本及其组合证明

Combinatorial degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma with a combinatorial proof

本文提出了广义Z_p-图克勒引理的组合度版本，并通过纯粹组合的方法证明了该定理，同时引入了一个独立的组合霍普夫追踪公式。

An Arithmetic Analogue of Fox's Triangle Removal Argument

We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group $\mathbb{F}_2^n$. A triangle in $\mathbb{F}_2^n$ is a triple $(x,y,z)$ such that $x+y+z = 0$. The triangle removal lemma for $\mathbb{F}_2^n$ states that for every $ε> 0$ there is a $δ> 0$, such that if a subset $A$ of $\mathbb{F}_2^n$ requires the removal of at least $ε\cdot 2^n$ elements to make it triangle-free, then it must contain at least $δ\cdot 2^{2n}$ triangles. This problem was first studied by Green [Gre05] who proved a lower bound on $δ$ using an arithmetic regularity lemma…

A connection between matchings and removal in abelian groups

In a finite abelian group $G$, define an additive matching to be a collection of triples $(x_i, y_i, z_i)$ such that $x_i + y_j + z_k = 0$ if and only if $i = j = k$. In the case that $G = \mathbb{F}_2^n$, Kleinberg, building on work of Croot-Lev-Pach and Ellenberg-Gijswijt, proved a polynomial upper bound on the size of an additive matching. Fox and Lovász used this to deduce polynomial bounds on Green's arithmetic removal lemma in $\mathbb{F}_2^n$. If $G$ is taken to be an arbitrary finite abelian group, the questions of bounding the size of an additive matching and giving bounds for Green's…

复杂度1的分划正则算术模式的诱导移除

Induced arithmetic removal for partition-regular patterns of complexity 1

本文证明了对于复杂度1的分划正则算术模式，可以在有限域向量空间中通过重新着色完全移除该模式，无需保留任何例外。

Partisan Combinatorial Game of Edge and Vertex Removal on Graphs

We consider three variants of a partisan combinatorial game between two players, Left and Right, played on an undirected simple graph. Left is able to delete vertices (and incident edges) while Right is able to delete edges. This natural extension of a similar impartial game gives a clear advantage to one player by allowing them the ability to play on a small subgraph which the other player can not. Our last variant removes this advantage by assuming a move is valid for one player if and only if the other player has a valid move on the same graph. In this case, we show that the ability to remo…

一个高效的非对称移除引理及其局限性

An efficient asymmetric removal lemma and its limitations

本文给出 Csaba 非对称三角形移除定理的无正则性证明，将 C₅ 副本数改进至最优 poly(ε)n⁵，并在 Ruzsa 猜想下证明并非所有无三角形三部图都 K₃-丰富。

Efficient Removal Lemmas for Matrices

The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following (ordered) matrix removal lemma: For any finite alphabet $Σ$, any hereditary property $\mathcal{P}$ of matrices over $Σ$, and any $ε> 0$, there exists $f_{\mathcal{P}}(ε)$ such that for any matrix $M$ over $Σ$ that is $ε$-far from satisfying $\mathcal{P}$, most of the $f_{\mathcal{P}}(ε) \times f_{\mathcal{P}}(ε)$ submatrices of $M$ do not satisfy $\math…

Lah numbers and Lindström's lemma

We provide a combinatorial interpretation of Lah numbers by means of planar networks. Henceforth, as a conesquence of Lindström's lemma, we conclude that the related Lah matrix possesses a remarkable property of total non-negativity.

利用洛瓦兹局部引理的形式化概率方法处理组合结构

Formal Probabilistic Methods for Combinatorial Structures using the Lovász Local Lemma

本文提出在 Isabelle/HOL 中基于 locale 的模块化框架，首次形式化洛瓦兹局部引理，用于形式化组合结构存在性的概率证明，并通过超图着色经典引理验证其适用性。

The bricklayer problem and the Strong Cycle Lemma

We introduce a combinatorial enumeration problem that is solved using generalized Catalan numbers. We also study generalizations of the Cycle Lemma beyond the computation of the generalized Catalan numbers.

An extended Schur's lemma and its application

The Springer modules have a combinatorial property called ``coincidence of dimensions,'' i.e., the Springer modules are naturally decomposed into submodules with common dimensions. Morita and Nakajima proved the property by giving modules with common dimensions whose induced modules are isomorphic to the submodules of Springer modules. They proved that the induced modules are isomorphic to the submodules, by showing the coincidence of their characters. Our aim is to construct isomorphisms between the induced modules and their corresponding submodules in a combinatorial manner. For this purpose…

具有Schatten范数和熵的稀疏正则方法

The sparse regularity method with Schatten norms and entropy

本文提出基于Schatten范数的稀疏图正则方法，导出k-圈移除引理，应用于加法组合学中的线性方程与多边形模式，并推广至稠密图移除引理。

Extensions of Sperner and Tucker's lemma for manifolds

The Sperner and Tucker lemmas are combinatorial analogous of the Brouwer and Borsuk - Ulam theorems with many useful applications. These classic lemmas are concerning labellings of triangulated discs and spheres. In this paper we show that discs and spheres can be substituted by large classes of manifolds with or without boundary.

Two Quick Proofs of a Catalan Lemma Needed by Lisa Sauermann and Yuval Wigderson

We present two proofs each, all shorter than the original proofs, of two elegant combinatorial identities that came up in the beautiful article arXiv:2010.00077 .

Deducing the Density Hales-Jewett Theorem from an infinitary removal lemma

We offer a new proof of Furstenberg and Katznelson's density version of the Hales-Jewett Theorem: For any $δ> 0$ there is some $N_0 \geq 1$ such that whenever $A \subseteq [k]^N$ with $N \geq N_0$ and $|A|\geq δk^N$, $A$ contains a \textbf{combinatorial line}: that is, for some $I \subseteq [N]$ nonempty and $w_0 \in [k]^{[N]\setminus I}$ we have A \supseteq \{w: w|_{[N]\setminus I} = w_0, w|_I = \rm{const.}\}. Following Furstenberg and Katznelson, we first show that this result is equivalent to a `multiple recurrence' assertion for a class of probability measures enjoying a certain kind of st…

Sperner's Lemma, Brouwer's fixed-point theorem, and cohomology

The proof of Brouwer's fixed-point theorem based on Sperner's lemma is often presented as an elementary combinatorial alternative to advanced proofs based on algebraic topology. The goal of this note is to show that: (i) the combinatorial proof of Sperner's Lemma can be considered as a cochain-level version, written in the combinatorial language, of a standard cohomological argument; (ii) the standard deduction of Brouwer's theorem from Sperner's lemma is similar to the usual deduction of Brouwer's theorem from the no-retraction theorem and is closely related to the notion of a simplicial appr…

Generalizations of Tucker-Fan-Shashkin lemmas

Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT-manifolds.

Schur Polynomials through Lindström Gessel Viennot Lemma

In this article, we use Lindström Gessel Viennot Lemma to give a short, combinatorial, visualizable proof of the identity of Schur polynomials -- the sum of monomials of Young tableaux equals to the quotient of determinants. As a by-product, we have a proof of Vandermonde determinant without words. We also prove the cauchy identity. In the remarks, we discuss factorial Schur polynomials, dual Cauchy identity and the relation beteen Newton interpolation formula.

A Generalized Turán Problem and its Applications

The investigation of conditions guaranteeing the appearance of cycles of certain lengths is one of the most well-studied topics in graph theory. In this paper we consider a problem of this type which asks, for fixed integers ${\ell}$ and $k$, how many copies of the $k$-cycle guarantee the appearance of an $\ell$-cycle? Extending previous results of Bollobás--Győri--Li and Alon--Shikhelman, we fully resolve this problem by giving tight (or nearly tight) bounds for all values of $\ell$ and $k$. We also present a somewhat surprising application of the above mentioned estimates to the study of the…

On linear configurations in subsets of compact abelian groups, and invariant measurable hypergraphs

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:…

An Infinitary version of Sperner's lemma

We prove an extension of the well-known combinatorial-topological lemma of E. Sperner to the case of infinite-dimensional cubes. It is obtained as a corollary to an infinitary extension of the Lebesgue Covering Dimension Theorem.