The counting lemma for regular <i>k</i>‐uniform hypergraphs

Nagle, Brendan; Rödl, Vojtěch; Schacht, Mathias

doi:10.1002/rsa.20117

Cited by 190 publications

(299 citation statements)

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“…In fact Szémeredi's regularity lemma, the main tool in the proof of his theorem, roughly speaking says that every graph can be decomposed into a few number of subgraphs such that most of them are quasi-random (we refer the reader to Tao's survey [35] for a precise formulation of the regularity lemma in terms of the h C4 (·) 1/4 norm). Recently Gowers [13,14] defined a hypergraph version of this norm, and subsequently he [12] and Nagle, Rödl, Schacht, and Skokan [22,26,25] independently established a hypergraph regularity lemma which easily implies Szemerédi's theorem in its full generality, and even stronger theorems such as Furstenberg-Katznelson's multi-dimensional arithmetic progression theorem [24,9], a result that the only known proof for it at the time was through ergodic theory [11]. In fact arithmetic version of the Gowers norm has interesting interpretations in ergodic theory, and has been studied from that aspect [19].…”

Section: Introductionmentioning

confidence: 99%

Graph norms and Sidorenko’s conjecture

Hatami

2010

Isr. J. Math.

169

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Let H and G be two finite graphs. Define hH (G) to be the number of homomorphisms from H to G. The function hH (•) extends in a natural way to a function from the set of symmetric matrices to R such that for AG, the adjacency matrix of a graph G, we have hH (AG) = hH (G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then hH (•) 1/m is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovász that asks for a characterization of graphs H for which the function hH (•) 1/m is a norm.We prove that hH (•) 1/m is a norm if and only if a Hölder type inequality holds for H. We use this inequality to prove both positive and negative results, showing that hH (•) 1/m is a norm for certain classes of graphs, and giving some necessary conditions on the structure of H when hH (•) 1/m is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact for such graphs we can prove statements that are much stronger than the assertion of Sidorenko's conjecture.We also investigate the hH (•) 1/m norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the 2n-th Schatten-von Neumann norms.

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Section: Introductionmentioning

confidence: 99%

Graph norms and Sidorenko’s conjecture

Hatami

2010

Isr. J. Math.

169

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“…The above arguments extend (with some nontrivial difficulty) to hypergraphs, and to proving Szemerédi's theorem for progressions of length k > 3; the k = 4 case was handled in [9], [10] (see also [20] for a more recent proof), and the general case in [33], [34], [32], [31] and [21] (see also [42], [45] for more recent proofs). We sketch the k = 4 arguments here (broadly following the ideas from [42], [45]).…”

Section: Then There Exists Functionsfmentioning

confidence: 99%

The ergodic and combinatorial approaches to Szemerédi’s theorem

Tao¹

2007

CRM Proceedings and Lecture Notes

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Abstract. A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graph-theoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourier-analytic approach of Gowers, and the hypergraph approach of Nagle-Rödl-Schacht-Skokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different. IntroductionThese lecture notes will be centred upon the following fundamental theorem of Szemerédi: This theorem is rather striking, because it assumes almost nothing on the given set A -other than that it is large -and concludes that A is necessarily structured in the sense that it contains arithmetic progressions of any given length k. This is a property special to arithmetic progressions (and a few other related patterns). Consider for instance the question asking whether a set A of positive density must contain a triplet of the form {x, y, x + y}. (Compare with the triplet {x, y, x+y 2 }, which is an arithmetic progression of length three.) It is then clear that the odd numbers, which are certainly a set of positive upper density, do not contain such triples (see however Theorem 6.1 below). Or for another example, consider whether a set of positive upper density must contain a pair {x, x + 2}. The multiples of 3 provide an immediate counterexample. (This is basically why the methods from [25] can leverage Szemerédi's theorem to show that the primes contain arbitrarily long arithmetic progressions, but are currently unable to make any progress whatsoever on the twin prime conjecture.) But the arithmetic progressions seem to be substantially more "indestructable" than these other types of patterns, in that they seem to occur in any large set A no matter how one tries to rearrange A to eliminate all the progressions.We have contrasted Szemerédi's theorem with some negative results where the selected pattern need not occur. Now let us give the opposite contrast, in which it becomes very 1991 Mathematics Subject Classification. 11N13, 11B25, 374A5. The author is supported by a grant from the Packard Foundation. Note that if we could just set a = b in these parallelograms then we could find infinitely progressions of length three. Alas, things are not so easy, and while progressions are certainly intimately related to parallelograms (and more generally to higher-dimensional parallelopipeds, for which an analogue of Proposition 1.2 can be easily located), the existence of the latter does not instantly imply the existence of the former without substantial additional effort. For example, one can easily modify Proposition 1.2 to locate, for any k ≥ 1, infinitely ma...

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“…This hypergraph has no copy of the tetrahedron K 3 4 but crossing triples would be gray in any 3-CRH that models it. Strong hypergraph regularity was developed in the 3-uniform case by Frankl and Rödl [26] and then for the general r-uniform case by Gowers [29], Rödl and Skokan [44,45] and Nagle, Rödl and Schacht [36]. In these formulations, the notion of how overlapping hyperedges interact is captured by structures known as complexes.…”

Section: Hypergraph Edit Distancementioning

confidence: 99%

The edit distance in graphs: Methods, results, and generalizations

Martin

2016

Recent Trends in Combinatorics

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The edit distance is a very simple and natural metric on the space of graphs. In the edit distance problem, we fix a hereditary property of graphs and compute the asymptotically largest edit distance of a graph from the property. This quantity is very difficult to compute directly but in many cases, it can be derived as the maximum of the edit distance function. Szemerédi's regularity lemma, stronglyregular graphs, constructions related to the Zarankiewicz problem -all these play a role in the computing of edit distance functions. The most powerful tool is derived from symmetrization, which we use to optimize quadratic programs that define the edit distance function. In this paper, we describe some of the most common tools used for computing the edit distance function, summarize the major current results, outline generalizations to other combinatorial structures, and pose some open problems.

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The counting lemma for regular k‐uniform hypergraphs

Cited by 190 publications

References 40 publications

Graph norms and Sidorenko’s conjecture

Graph norms and Sidorenko’s conjecture

The ergodic and combinatorial approaches to Szemerédi’s theorem

The edit distance in graphs: Methods, results, and generalizations

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