A short proof of Gowers’ lower bound for the regularity lemma

Moshkovitz, Guy; Shapira, A.

doi:10.1007/s00493-014-3166-4

Cited by 23 publications

(30 citation statements)

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“…This allowed us to obtain a new proof of Fox's Ack 2 (log 1/ǫ) upper bound for the graph removal lemma [6] (since the stronger notion allows to count small subgraphs). We believe that it should be possible to match our lower bounds with Ack k (log 1/p) upper bounds (even for a slightly stronger notion analogous to the one used in [18]). We think that it should be possible to deduce from such an upper bound an Ack k (log 1/ǫ) upper bound for the k-graph removal lemma.…”

Section: Barriers and Other Aspects Of The Main Proofmentioning

confidence: 96%

“…Roughly speaking, we will show that for a k-graph with pn k edges, every δ -regular partition has order at least Ack k (log 1/p). In a recent paper [18] we proved that in graphs, one can prove a matching Ack 2 (log 1/p) upper bound, even for a slightly stronger notion than δ -regularity. This allowed us to obtain a new proof of Fox's Ack 2 (log 1/ǫ) upper bound for the graph removal lemma [6] (since the stronger notion allows to count small subgraphs).…”

Section: Barriers and Other Aspects Of The Main Proofmentioning

confidence: 99%

“…For the proof of Claim 4.8 we will need the following lemma from [18] (which improved upon a similar lemma from [10]). Lemma 4.9 (restatement of Lemma 2.3 in [18]). Let G be a bipartite graph on (X, Y) that is 1 16 -balanced.…”

Section: Core Constructionmentioning

confidence: 99%

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A Tight Bound for Hyperaph Regularity

Moshkovitz

Shapira

2019

Geom. Funct. Anal.

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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 ≥ 2, thus confirming a prediction of Tao. Prior to our work, the only result of this type was Gowers' famous lower bound for graph regularity. IntroductionAs part of the proof of his eponymous theorem [35] on arithmetic progressions in dense sets of integers, Szemerédi developed (a variant of what is now known as) the graph regularity lemma [36]. The lemma roughly states that the vertex set of every graph can be partitioned into a bounded number of parts such that almost all the bipartite graphs induced by pairs of parts in the partition are quasi-random. In the past four decades this lemma has become one of the (if not the) most powerful tools in extremal combinatorics, with applications in many other areas of mathematics. We refer the reader to [17,29] for more background on the graph regularity lemma, its many variants and its numerous applications.Perhaps the most important and well-known application of the graph regularity lemma is the original proof of the triangle removal lemma, which states that if an n-vertex graph G contains only o(n 3 ) triangles, then one can turn G into a triangle-free graph by removing only o(n 2 ) edges (see [3] for more details). It was famously observed by Ruzsa and Szemerédi [32] that the triangle removal lemma implies Roth's theorem [31], the special case of Szemerédi's theorem for 3-term arithmetic progressions. The problem of extending the triangle removal lemma to the hypergraph setting was raised by Erdős, Frankl and Rödl [5]. One of the main motivations for obtaining such a result was the observation of Frankl and Rödl [8] (see also [34]) that such a result would allow one to extend the Ruzsa-Szemerédi [32] argument and thus obtain an alternative proof of Szemerédi's theorem for progressions of arbitrary length.The quest for a hypergraph regularity lemma, which would allow one to prove a hypergraph removal lemma, took about 20 years. The first milestone was the result of Frankl and Rödl [8], who obtained a regularity lemma for 3-uniform hypergraphs. About 10 years later, the approach of [8] was extended to hypergraphs of arbitrary uniformity by Rödl, Skokan, Nagle and Schacht [22,30]. At the same time, Gowers [13] obtained an alternative version of the regularity lemma for k-uniform hypergraphs (from now on we will use k-graphs instead of k-uniform hypergraphs). Shortly after, Tao [38] and Rödl and Schacht [27,28] obtained two more versions of the lemma.As it turned out, the main difficulty with obtaining a regularity lemma for k-graphs was...

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Section: Barriers and Other Aspects Of The Main Proofmentioning

confidence: 96%

Section: Barriers and Other Aspects Of The Main Proofmentioning

confidence: 99%

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A Tight Bound for Hyperaph Regularity

Moshkovitz

Shapira

2019

Geom. Funct. Anal.

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“…Our proof will critically rely on the following lemma from [12], which improved upon a similar lemma from [6].…”

Section: Preliminary Lemmasmentioning

confidence: 99%

A sparse regular approximation lemma

Moshkovitz

Shapira

2019

Trans. Amer. Math. Soc.

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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 ǫ, δ, where we think of δ as a constant. The goal is to construct an ǫ-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 raises the following question: can one obtain quantitative bounds for SRAL that are significantly better than those associated with the regularity lemma?Our first result answers the above question affirmatively by proving an upper bound for SRAL given by a tower of height O(log 1/p). This allows us to reprove Fox's upper bound for the graph removal lemma. Our second result is a matching lower bound for SRAL showing that a tower of height Ω(log 1/p) is unavoidable. We in fact prove a more general multicolored lower bound which is essential for proving lower bounds for the hypergraph regularity lemma.

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“…Gowers [25] used a probabilistic construction to show that such an enormous bound is indeed necessary. Consult [15], [34], [20] for other proofs that improve on various aspects of the result. Szemerédi's regularity lemma was extended to k-uniform hypergraphs by Gowers [24,26] and by Nagle, Rödl, Schacht, and Skokan [35].…”

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confidence: 99%

Density and regularity theorems for semi-algebraic hypergraphs

Fox¹,

Pach²,

Suk³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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A k-uniform semi-algebraic hypergraph H is a pair (P, E), where P is a subset of R d and E is a collection of k-tuples {p 1 , . . . , p k } ⊂ P such that (p 1 , . . . , p k ) ∈ E if and only if the kd coordinates of the p i -s satisfy a boolean combination of a finite number of polynomial inequalities. The complexity of H can be measured by the number and the degrees of these inequalities and the number of variables (coordinates) kd. Several classical results in extremal hypergraph theory can be substantially improved when restricted to semialgebraic hypergraphs.Substantially improving a theorem of Fox, Gromov, Lafforgue, Naor, and Pach, we establish the following "polynomial regularity lemma": For any 0 < ε < 1/2, the vertex set of every k-uniform semi-algebraic hypergraph H = (P, E) can be partitioned into at most (1/ε) c parts P 1 , P 2 , . . ., as equal as possible, such that all but at most an ε-fraction of the k-tuples of parts (P i1 , . . . , P ik ) are homogeneous in the sense that either every k-tuple (p i1 , . . . , p ik ) ∈ P i1 × . . . × P ik belongs to E or none of them do. Here c > 0 is a constant that depends on the complexity of H. We also establish an improved lower bound, single exponentially decreasing in k, on the best constant δ > 0 such that the vertex classes P 1 , . . . , P k of every k-partite k-uniform semi-algebraic hypergraph Any disjoint finite sets P 1 , . . . ,with the property that every k-tuple formed by taking one point from each P i has the same order type.The above techniques carry over to property testing. We show that for any typical hereditary hypergraph property Q, there is a randomized algorithm with query complexity (1/ε) c(Q) to determine (with probability at least .99) whether a k-uniform semi-algebraic hypergraph H = (P, E) with constant description complexity is ε-near to having property Q, that is, whether one can change at most ε|P | k hyperedges of H in order to obtain a hypergraph that has the property. The testability of such properties for general k-uniform hypergraphs was first shown by Alon and Shapira (for graphs) and by Rödl and Schacht (for k > 2). The query complexity time of their algorithms is enormous, growing considerably faster than a tower function.

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A short proof of Gowers’ lower bound for the regularity lemma

Cited by 23 publications

References 3 publications

A Tight Bound for Hyperaph Regularity

A Tight Bound for Hyperaph Regularity

A sparse regular approximation lemma

Density and regularity theorems for semi-algebraic hypergraphs

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