Yoshiyasu Ishigami
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.
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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 a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding open problem of Erdős and Hajnal in Ramsey theory. We further extend this result to prove tight bounds on various other hypergraph Ramsey numbers.
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.
Low discrepancy and spectral gap for directed hypergraphs via spectral regularity lemma for tensors
本文证明有向均匀超图的低差异等价于谱间隙，通过建立基于变分特征值的Frieze-Kannan型张量谱正则引理，给出Lenz-Mubayi定理的简短证明，并讨论相关变体。
The number of cliques in hypergraphs with forbidden subgraphs
本文研究禁用完全 r 部超图的 $(r-1)$-一致超图中 r 团的最大数量，借助超图移除引理证明上界，并用超图 Turán 数给出下界。
A result due to Gyárfás, Hubenko, and Solymosi (answering a question of Erdös) states that if a graph $G$ on $n$ vertices does not contain $K_{2,2}$ as an induced subgraph yet has at least $c\binom{n}{2}$ edges, then $G$ has a complete subgraph on at least $\frac{c^2}{10}n$ vertices. In this paper we suggest a "higher-dimensional" analogue of the notion of an induced $K_{2,2}$ which allows us to generalize their result to $k$-uniform hypergraphs. Our result also has an interesting consequence in discrete geometry. In particular, it implies that the fractional Helly theorem can be derived as a …
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…
A \emph{simple} matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say that a (0,1)-matrix $A$ has $F$ as a \emph{Berge hypergraph} if there is a submatrix $B$ of $A$ and some row and column permutation of $F$, say $G$, with $G\le B$. Letting $||A||$ denote the number of columns in $A$, we define the extremal function $Bh(m,{ F})=\max\{||A||\,:\, A \hbox{ is }m\hbox{-rowed simple matrix with no Berge hypergraph }F\}$. We determine the asymptotics of $Bh(m,F)$ for all $3$- and $4$-rowed $F$ and most $5$-rowed $F$. For certain $F$, this becomes the problem of determini…
Towards an optimal hypergraph container lemma
本文提出并证明了超图容器引理的两个新版本，具有简洁证明、替代几乎独立性概念及改善容器数量对均匀性依赖等特征，达到二阶矩方法的自然障碍。
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 well-known Erdős-Hajnal conjecture states that for any graph $F$, there exists $ε>0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^ε$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every …
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…
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.
For $n\geq 3$ and $r=r(n) \geq 3$, let $\boldsymbol{k} =\boldsymbol{k}(n)=(k_1, \ldots, k_n)$ be a sequence of non-negative integers with sum $M(\boldsymbol{k})=\sum_{j=1}^{n} k_j$. We assume that $M(\boldsymbol{k})$ is divisible by $r$ for infinitely many values of $n$, and restrict our attention to these values. Let $X=X(n)$ be a simple $r$-uniform hypergraph on the vertex set $V=\{v_1,v_2, \ldots, v_n\}$ with $t$ edges and maximum degree $x_{\max}$. We denote by $\mathcal{H}_r(\boldsymbol{k})$ the set of all simple $r$-uniform hypergraphs on the vertex set $V$ with degree sequence $\boldsym…
We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about the connection of the largest cardinalities (or weight) of structures satisfying some hereditary property and substructures satisfying the same hereditary property. We use it to show how results concerning forbidden subposet problems in the Boolean poset imply analogous results in the poset of subspaces of a finite vector space. We also study generalized forbidden subposet problems in the poset of subspaces.
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-…
We give a new approach to handling hypergraph regularity. This approach allows for vertex-by-vertex embedding into regular partitions of hypergraphs, and generalises to regular partitions of sparse hypergraphs. We also prove a corresponding sparse hypergraph regularity lemma.
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。
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…
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…
A {\it $k$-uniform hypergraph} $\mathcal{H}=(V, E)$ consists of a set $V$ of vertices and a set $E$ of hyperedges ($k$-hyperedges), which is a family of $k$-subsets of $V$. A {\it forbidden $k$-homogeneous (or forbidden $k$-hypergraph)} access structure $\mathcal{A}$ is represented by a $k$-uniform hypergraph $\mathcal{H}=(V, E)$ and has the following property: a set of vertices (participants) can reconstruct the secret value from their shares in the secret sharing scheme if they are connected by a $k$-hyperedge or their size is at least $k+1$. A forbidden $k$-homogeneous access structure has …
Large cliques or co-cliques in hypergraphs with forbidden order-size pairs
该研究探讨超图中Erdős-Hajnal问题变体，针对禁止特定阶-大小对超图的情形，给出三元系m=4时齐次集大小界并确定其增长率类型。
Characterizing unimodular laminar hypergraphs via forbidden subhypergraphs
本文通过禁止奇环、鳄梨和树屋结构，刻画了层状超图的全单模性，并解决了Padberg猜想的一个特例，应用于整数规划。
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.
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 …
Venn diagrams as forbidden hypergraph traces
研究不含\(k\)维恩图作为迹的集系最大规模，证明\(\text{ex}_{tr}(n,VD_k)=O_k(n^{2^k - 2k + 1})\)改进了界，对\(k\)归纳证明，还给出相关下界构造、显式界及固定均匀性迹结果。
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
研究在NIP强k-远结构中可定义的(k + 1)-一致超图，通过引入k-强诚实定义，证明其满足齐次正则引理，扩展了Chernikov和Starchenko的相关成果，将强诚实定义理论推广到高元情形。
We establish a so-called counting lemma that allows embeddings of certain linear uniform hypergraphs into sparse pseudorandom hypergraphs, generalizing a result for graphs [Embedding graphs with bounded degree in sparse pseudorandom graphs, Israel J. Math. 139 (2004), 93-137]. Applications of our result are presented in the companion paper [Counting results for sparse pseudorandom hypergraphs II].
In this work, we establish an analogue result of the Erdös-Stone theorem of weighted digraphs using Regularity Lemma of digraphs. We give a stability result of oriented graphs and digraphs with forbidden blow-up transitive triangle and show that almost all oriented graphs and almost all digraphs with forbidden blow-up transitive triangle are almost bipartite respectively.
Perfect stable regularity lemma and slice-wise stable hypergraphs
本文研究了超图的模型论稳定性及其对顶点划分的正则性引理加强，给出了三元情形的完整分类，并构造反例回应了Terry和Wolf的猜想，结果适用于有限图超积与可定义Keisler测度。
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…
Regularity for hypergraphs with bounded VC$_2$ dimension
本文针对有界VC$_2$维数的3-图，证明了可将其正则性界限在阿克曼层级中降低一层且达到最佳，核心方法是基于柱体正则性引理设计新方法并扩展至超图。
Finding an almost perfect matching in a hypergraph avoiding forbidden submatchings
该研究证明了Glock等人关于高围长$(n,q,r)$-Steiner系统存在性猜想的近似版本，推广了超图匹配相关经典定理，具有多项应用价值。
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.
The singleton hypergraph is extremal for the Isolation Lemma
研究具有\(n\)个顶点的无包含关系超图的隔离权重分配，证明其数量至少为\(n\sum_{j = 0}^{d - 1}j^{n - 1}\)，单元素边超图能达到此界，还证明了更一般边权重目标类的界，证实了相关猜想。
Polynomial-in-$r$ bounds for forbidden traces of uniform hypergraphs
本文提出一个将禁用迹的固定一致性界转化为对一致性多项式依赖界的一般原理，并给出迹-$C_4$-自由超图的上下界，上界为$O(r^{3/2+\varepsilon}m^{3/2})$，下界为$\Omega(r^{1/2}m^{3/2})$。
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.
Ramsey properties of randomly perturbed hypergraphs
本文研究随机扰动 3-一致超图的 Ramsey 性质，利用融合强弱形式的新超图正则性引理及元组引理，确定了诱导扩展团两色 Ramsey 性质所需随机超边的阈值。
We formulate and prove a general result in spirit of hypergraph removal lemma for measurable functions of several variables.
Let F be a hypergraph and let forb(m,F) denote the maximum number of edges a hypergraph with m vertices can have if it doesn't contain F as a subhypergraph. A conjecture of Anstee and Sali predicts the asymptotic behaviour of forb(m,F) for fixed F. In this paper we prove that even finding this predicted asymptotic behaviour is an NP-hard problem, meaning that if the Anstee-Sali conjecture were true, finding the asymptotics of forb(m,F) would be NP-hard.
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:…
Szemerédi's regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemerédi's theorem on arithmetic progressions . In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a variant of this lemma which introduces a new parameter $F$. This stronger version of the regularity lemma was iterated in a recent paper of the author to reprove the analogous regularity lemma for hypergraphs.
We describe a method that we believe may be foundational for a comprehensive theory of generalised Turan problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that any 3-graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a …
Motivated by his work on the classification of countable homogeneous oriented graphs, Cherlin asked about the typical structure of oriented graphs (i) without a transitive triangle, or (ii) without an oriented triangle. We give an answer to these questions (which is not quite the predicted one). Our approach is based on the recent `hypergraph containers' method, developed independently by Saxton and Thomason as well as by Balogh, Morris and Samotij. Moreover, our results generalise to forbidden transitive tournaments and forbidden oriented cycles of any order, and also apply to digraphs. Along…
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。
Binary clustering systems are closely related to monotone transit functions. An interesting class are pyramidal transit functions defined by the fact that their transit sets form an interval hypergraph. We investigate here properties of transit function $R$, such as union-closure, that are sufficient to ensure that $R$ is at least weakly pyramidal. Necessary conditions for pyramidal transit functions are derived from the five forbidden configurations in Tucker's characterization of interval hypergraphs. The first corresponds to $β$-acyclicity, also known as total balancedness, for which we obt…
Spanning spheres in Dirac hypergraphs
本文在无孤立顶点且每个 (k−1) 元组至少属于 n/2+o(n) 条边的条件下，证明 k-一致超图含与 (k−1) 维球同胚的生成子图，渐近证实相关猜想，并用完全 blow-up 覆盖框架替代吸收法等传统工具。
Multilevel polynomial partitioning and semialgebraic hypergraphs: regularity, Turán, and Zarankiewicz results
本研究提出多层多项式划分方案，改进了半代数超图的正则引理、Turán型与Zarankiewicz型结果，并给出多个相关领域应用。
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…
We show that for every $k \in \mathbb{N}$ there exists $C > 0$ such that if $p^k \ge C \log^8 n / n$ then asymptotically almost surely the random graph $G_{n,p}$ contains the $k$\textsuperscript{th} power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kühn and Osthus. Moreover, our proof provides a randomized quasi-polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi-polynomial algorithm for finding a tight Hamilton cycle in the random…
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