## Szemerédi’s regularity lemma revisited

Venue: | Contrib. Discrete Math |

Citations: | 15 - 3 self |

### BibTeX

@ARTICLE{Tao_szemerédi’sregularity,

author = {Terence Tao},

title = {Szemerédi’s regularity lemma revisited},

journal = {Contrib. Discrete Math},

year = {},

pages = {2006}

}

### OpenURL

### Abstract

Abstract. 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 [19], [18]. In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a slightly stronger variant of this lemma, related to similar strengthenings of that lemma in [1]. This stronger version of the regularity lemma was extended in [21] to reprove the analogous regularity lemma for hypergraphs. 1.

### Citations

256 | On sets of integers containing no k elements in arithmetic progression, Acta Arith.27
- Szemerédi
- 1975
(Show Context)
Citation Context ...tract. 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 =-=[19]-=-, [18]. In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a slightly stronger variant of this lemma, related to ... |

217 | Szemerédi’s Regularity Lemma and its applications in graph theory
- Komós, Simonovits
- 1993
(Show Context)
Citation Context ...d independently of the number of vertices in the original graph. The regularity lemma has had many applications in graph theory, computer science, discrete geometry and in additive combinatorics, see =-=[10]-=- for a survey. In particular, this lemma and its variants play an important role in Szemerédi’s celebrated theorem [19] that any subset of the integers of positive density contain arbitrarily long ari... |

167 |
Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions
- Furstenberg
- 1977
(Show Context)
Citation Context ...her as a structure theorem for events or random variables in a product probability space. This change of perspective is analogous to Furstenberg’s highly successful approach to Szemerédi’s theorem in =-=[6]-=-, in which the purely combinatorial result of Szemerédi was recast as a statement about recurrence for arbitrary events or random variables in a probability-preserving system. Just as Furstenberg’s ch... |

163 | Efficient Testing of Large Graphs
- Alon, Fischer, et al.
- 1999
(Show Context)
Citation Context ...ma from the perspective of probability theory and information theory instead of graph theory, and observe a slightly stronger variant of this lemma, related to similar strengthenings of that lemma in =-=[1]-=-. This stronger version of the regularity lemma was extended in [21] to reprove the analogous regularity lemma for hypergraphs. 1. Introduction Szemerédi’s regularity lemma, introduced by Szemerédi in... |

128 | Regular partitions of graphs - Szemerédi - 1978 |

100 |
Hypergraph regularity and the multidimensional szemerédi theorem. submitted
- Gowers
(Show Context)
Citation Context ...t the primes contained arbitrarily long arithmetic progressions. The lemma has also had a number of generalizations to hypergraphs of varying degrees of strength, see [3], [4], [5], [13], [14], [15], =-=[9]-=-, [21]. The more recent formulations of the hypergraph lemma are in fact strong enough to rather easily imply Szemerédi’s theorem on arithmetic progressions, as well as a multidimensional version due ... |

78 | An ergodic Szemerédi theorem for commuting transformations
- Furstenberg, Katznelson
- 1979
(Show Context)
Citation Context ...nuscript. The author is also indebted to the anonymous referees for many useful suggestions and corrections. The author is supported by a grant from the Packard Foundation. 12 TERENCE TAO Katznelson =-=[7]-=-. They were also used in the recent paper [22] establishing infinitely many constellations of any given shape in the Gaussian primes. The proof of Szemerédi’s lemma is now standard in the literature. ... |

71 | The counting lemma for regular k-uniform hypergraphs. Random Structures Algorithms
- Nagle, Rödl, et al.
(Show Context)
Citation Context ...howing in [11] that the primes contained arbitrarily long arithmetic progressions. The lemma has also had a number of generalizations to hypergraphs of varying degrees of strength, see [3], [4], [5], =-=[13]-=-, [14], [15], [9], [21]. The more recent formulations of the hypergraph lemma are in fact strong enough to rather easily imply Szemerédi’s theorem on arithmetic progressions, as well as a multidimensi... |

70 | Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms 25
- Rödl, Skokan
- 2004
(Show Context)
Citation Context ...1] that the primes contained arbitrarily long arithmetic progressions. The lemma has also had a number of generalizations to hypergraphs of varying degrees of strength, see [3], [4], [5], [13], [14], =-=[15]-=-, [9], [21]. The more recent formulations of the hypergraph lemma are in fact strong enough to rather easily imply Szemerédi’s theorem on arithmetic progressions, as well as a multidimensional version... |

64 |
Lower bounds of tower type for Szemerédi’s uniformity lemma
- Gowers
- 1997
(Show Context)
Citation Context ... deceptive, as it conceals the fact that J can in fact be extremely large depending on 1/ε, indeed there are examples where J grows like an exponential tower of height equal to some power of 1/ε (see =-=[8]-=-). However, the key point is that the bound on J does not depend on the cardinality of V1 or V2. Indeed we shall shortly give a probabilistic formulation in which V1 and V2 could be infinite (cf. [12]... |

57 |
On sets of integers containing no four elements in arithmetic progression
- Szemerédi
- 1969
(Show Context)
Citation Context ... 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 [19], =-=[18]-=-. In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a slightly stronger variant of this lemma, related to simila... |

44 | Every monotone graph property is testable - Alon, Shapira - 2005 |

44 |
Regularity lemmas for hypergraphs and quasi-randomness, Random Structures Algorithms 2
- Chung
- 1991
(Show Context)
Citation Context ...so crucial in showing in [11] that the primes contained arbitrarily long arithmetic progressions. The lemma has also had a number of generalizations to hypergraphs of varying degrees of strength, see =-=[3]-=-, [4], [5], [13], [14], [15], [9], [21]. The more recent formulations of the hypergraph lemma are in fact strong enough to rather easily imply Szemerédi’s theorem on arithmetic progressions, as well a... |

39 |
Extremal problems on set systems, Random Structures Algorithms 20
- Frankl, Rödl
- 2002
(Show Context)
Citation Context ... in showing in [11] that the primes contained arbitrarily long arithmetic progressions. The lemma has also had a number of generalizations to hypergraphs of varying degrees of strength, see [3], [4], =-=[5]-=-, [13], [14], [15], [9], [21]. The more recent formulations of the hypergraph lemma are in fact strong enough to rather easily imply Szemerédi’s theorem on arithmetic progressions, as well as a multid... |

35 | Applications of the regularity lemma for uniform hypergraphs
- Rödl, Skokan
(Show Context)
Citation Context ...also crucial showing in [9] that the primes contained arbitrarily long arithmetic progressions. The lemma has also had a number of generalizations to hypergraphs, see [1], [2], [3], [10], [11], [12], =-=[13]-=-, [7], [18]; these generalizations can rather easily imply Szemerédi’s theorem on arithmetic progressions, as well as a multidimensional version due to Furstenberg and Katznelson [5]. They were also u... |

23 |
Quasi-random hypergraphs, Random Structures Algorithms 1
- Chung, Graham
- 1990
(Show Context)
Citation Context ...rom ideas in ergodic theory) was also crucial showing in [9] that the primes contained arbitrarily long arithmetic progressions. The lemma has also had a number of generalizations to hypergraphs, see =-=[1]-=-, [2], [3], [10], [11], [12], [13], [7], [18]; these generalizations can rather easily imply Szemerédi’s theorem on arithmetic progressions, as well as a multidimensional version due to Furstenberg an... |

21 | The Gaussian primes contain arbitrarily shaped constellations - Tao |

12 |
The uniformity lemma for hypergraphs, Graphs Combin
- Frankl, Rödl
- 1992
(Show Context)
Citation Context ...ucial in showing in [11] that the primes contained arbitrarily long arithmetic progressions. The lemma has also had a number of generalizations to hypergraphs of varying degrees of strength, see [3], =-=[4]-=-, [5], [13], [14], [15], [9], [21]. The more recent formulations of the hypergraph lemma are in fact strong enough to rather easily imply Szemerédi’s theorem on arithmetic progressions, as well as a m... |

12 |
Note on a generalization of Roth’s theorem,Discrete and computational geometry, 825–827, Algorithms Combin. 25
- Solymosi
- 2003
(Show Context)
Citation Context ...to be strong enough for applications such as Szemerédi’s theorem or the FurstenbergKatznelson theorem 1 , except when concerning progressions or constellations consisting of at most three points (see =-=[17]-=-). In this paper we shall present a slightly different way of looking at Szemerédi’s regularity lemma, which we used in [21] to obtain a hypergraph regularity lemma with sufficient strength for applic... |

9 |
A variant of the hypergraph removal lemma, preprint
- Tao
(Show Context)
Citation Context ...y instead of graph theory, and observe a slightly stronger variant of this lemma, related to similar strengthenings of that lemma in [1]. This stronger version of the regularity lemma was extended in =-=[21]-=- to reprove the analogous regularity lemma for hypergraphs. 1. Introduction Szemerédi’s regularity lemma, introduced by Szemerédi in [19], is a fundamental tool in graph theory, and more precisely in ... |

8 |
Regular partitions of hypergraphs, preprint
- Rödl, Schacht
(Show Context)
Citation Context ... in [11] that the primes contained arbitrarily long arithmetic progressions. The lemma has also had a number of generalizations to hypergraphs of varying degrees of strength, see [3], [4], [5], [13], =-=[14]-=-, [15], [9], [21]. The more recent formulations of the hypergraph lemma are in fact strong enough to rather easily imply Szemerédi’s theorem on arithmetic progressions, as well as a multidimensional v... |

3 |
Szemerédi’s regularity lemma for the analyst, preprint
- Lovász, Szegedy
(Show Context)
Citation Context ...then in the random or pseudorandom case the event E will be almost completely uncorrelated 3 Note added in proof: a closely related version of this lemma was recently introduced in [1], [2]. See also =-=[12]-=- for yet another perspective on the regularity lemma, this time from functional analysis.4 TERENCE TAO with the events A1,A2. This corresponds to the well-known fact that when G is a random or pseudo... |

1 |
The primes contain arbitrarily long proper arithmetic progressions, preprint
- Green, Tao
(Show Context)
Citation Context ...integers of positive density contain arbitrarily long arithmetic progressions. A variant of this structure theorem (also borrowing heavily from ideas in ergodic theory) was also crucial in showing in =-=[11]-=- that the primes contained arbitrarily long arithmetic progressions. The lemma has also had a number of generalizations to hypergraphs of varying degrees of strength, see [3], [4], [5], [13], [14], [1... |