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

Citations: | 20 - 5 self |

### BibTeX

@MISC{Tao_acorrespondence,

author = {Terence Tao},

title = {A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma, preprint},

year = {}

}

### OpenURL

### Abstract

Abstract. 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 progressions. 1.

### Citations

225 | Ergodic Theory and Combinatorial Number Theory - Furstenberg - 1981 |

201 | Szemerédi’s regularity lemma and its applications in graph theory - Komlós, Simonovits - 1993 |

150 | The primes contain arbitrarily long arithmetic progressions - Green, Tao |

143 | Regular partitions of graphs, Problèmes combinatoires et théorie des graphes - Szemerédi - 1976 |

138 |
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(Show Context)
Citation Context ...ong arithmetic progressions. In 1977, Furstenberg obtained a new proof of Szemerédi’s theorem by deducing it from the following result in ergodic theory. Theorem 2.2 (Furstenberg recurrence theorem). =-=[7]-=-, [10] Let (Ω, Bmax,P) be a probability space (see Appendix A for probabilistic notation). Let T : Ω → Ω be a bi-measurable map which is probability preserving, thus P(T n A) = P(A) for all events A ∈... |

94 |
Hypergraph regularity and the multidimensional Szemerédi theorem, manuscript
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(Show Context)
Citation Context ...e of the triangle removal lemma of Ruzsa and Szemerédi [24], as well as the substantially more difficult hypergraph removal lemma of Nagle, Rödl, Schacht, and Skokan [19], [20], [22], [23] and Gowers =-=[12]-=- (as well as a later refinement in [30]). As this lemma is already strong enough to deduce Szemerédi’s theorem on arithmetic progressions (as well as a multidimensional generalisation due to Furstenbe... |

90 | A characterization of the (natural) graph properties testable with one-sided error - Alon, Shapira |

85 |
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- 1976
(Show Context)
Citation Context ...up proofs of certain statements which previously could only be proven via a regularity lemma. In particular, we will give an infinitary proof here of the triangle removal lemma of Ruzsa and Szemerédi =-=[24]-=-, as well as the substantially more difficult hypergraph removal lemma of Nagle, Rödl, Schacht, and Skokan [19], [20], [22], [23] and Gowers [12] (as well as a later refinement in [30]). As this lemma... |

78 |
Nonconventional ergodic averages and nilmanifolds
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(Show Context)
Citation Context ...s contained arbitrarily long progressions; this argument was almost entirely finitary in nature, yet at the same time it relied heavily on ideas from the infinitary world of ergodic theory (see [17], =-=[15]-=- for further discussion of this connection). In this paper we investigate a transference in the other direction, taking results from the finitary world of combinatorics (and in particular graph theory... |

75 | An ergodic Szemerédi theorem for commuting transformations
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- 1978
(Show Context)
Citation Context ...er refinement in [30]). As this lemma is already strong enough to deduce Szemerédi’s theorem on arithmetic progressions (as well as a multidimensional generalisation due to Furstenberg and Katznelson =-=[9]-=-), we have thus presented yet another proof of Szemerédi’s theorem here. These lemmas have some further applications; for theorem via topological dynamics, see [11]. However such methods do not seem t... |

67 | The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs Combin - Erdős, Frankl, et al. - 1986 |

67 | Regularity lemma for k-uniform hypergraphs, Random Structures and Algorithms
- Rödl, Skokan
(Show Context)
Citation Context ...n infinitary proof here of the triangle removal lemma of Ruzsa and Szemerédi [24], as well as the substantially more difficult hypergraph removal lemma of Nagle, Rödl, Schacht, and Skokan [19], [20], =-=[22]-=-, [23] and Gowers [12] (as well as a later refinement in [30]). As this lemma is already strong enough to deduce Szemerédi’s theorem on arithmetic progressions (as well as a multidimensional generalis... |

66 | The counting lemma for regular k-uniform hypergraphs
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(Show Context)
Citation Context ... will give an infinitary proof here of the triangle removal lemma of Ruzsa and Szemerédi [24], as well as the substantially more difficult hypergraph removal lemma of Nagle, Rödl, Schacht, and Skokan =-=[19]-=-, [20], [22], [23] and Gowers [12] (as well as a later refinement in [30]). As this lemma is already strong enough to deduce Szemerédi’s theorem on arithmetic progressions (as well as a multidimension... |

51 |
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(Show Context)
Citation Context ...ts such as Szemerédi’s theorem with recurrence results in ergodic theory. Let us recall Szemerédi’s theorem in a quantitative (finitary) form: Theorem 2.1 (Szemerédi’s theorem, quantitative version). =-=[26]-=- Let 0 < δ ≤ 1 and k ≥ 1. Let A be a subset of a cyclic group ZN := Z/NZ whose cardinality |A| is at least δN. Then there exist at least c(k, δ)N 2 pairs (x, r) ∈ ZN × ZN such that x, x + r, . . . , x... |

42 |
On certain sets of positive density
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(Show Context)
Citation Context ...onal (infinitary) form, which asserts that every set of integers of positive upper density contains arbitrarily long progressions. The converse implication also follows from an argument of Varnavides =-=[33]-=-. This particular formulation of Szemerédi’s theorem played an important role in the recent result [13] that the primes contained arbitrarily long arithmetic progressions. In 1977, Furstenberg obtaine... |

35 |
Extremal problems on set systems, Random Structure and Algorithms 20(2
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- 2002
(Show Context)
Citation Context ...h regularity lemma, which has only been satisfactorily achievedsCORRESPONDENCE BETWEEN GRAPH THEORY AND PROBABILITY THEORY 25 quite recently [19], [20], [12], [30] (with preliminary work in [5], [3], =-=[6]-=-). In the author’s view this is because the regularity lemma is a relatively “soft” component of the theory; in the infinitary framework, the “hard” components of the theory are now isolated in the th... |

33 | Applications of the regularity lemma for uniform hypergraphs, submitted
- Rödl, Skokan
(Show Context)
Citation Context ... theorem has a number of applications, most notably in giving a proof not only of Szemerédi’s theorem (Theorem 2.1) but also a multidimensional version due to Furstenberg and Katznelson [9]; see e.g. =-=[23]-=-, [12], [31] for further discussion of this connection, and [21] for some more applications of this theorem. A variant of this theorem was also used in [31] to establish that the Gaussian primes conta... |

33 | A quantitative ergodic theory proof of Szemerédi’s theorem
- Tao
- 2006
(Show Context)
Citation Context ...tary ones; this was for instance carried out for the Furstenberg-Weiss infinitary proof of van der Waerden’s 1s2 TERENCE TAO can be transferred to the finitary world, and vice versa; see for instance =-=[29]-=- for a finitary version of the infinitary ergodic approach to Szemerédi’s theorem. Such a fusion of ideas from both sources proved to be particularly crucial in the recent result [13] that the primes ... |

25 | The ergodic theoretical proof of Szemerédi’s theorem
- Furstenberg, Katznelson, et al.
- 1982
(Show Context)
Citation Context ...rithmetic progressions. In 1977, Furstenberg obtained a new proof of Szemerédi’s theorem by deducing it from the following result in ergodic theory. Theorem 2.2 (Furstenberg recurrence theorem). [7], =-=[10]-=- Let (Ω, Bmax,P) be a probability space (see Appendix A for probabilistic notation). Let T : Ω → Ω be a bi-measurable map which is probability preserving, thus P(T n A) = P(A) for all events A ∈ Bmax ... |

19 |
Quasi-random hypergraphs, Random Structures and Algorithms
- Chung, Graham
- 1990
(Show Context)
Citation Context ...rgraph regularity lemma, which has only been satisfactorily achievedsCORRESPONDENCE BETWEEN GRAPH THEORY AND PROBABILITY THEORY 25 quite recently [19], [20], [12], [30] (with preliminary work in [5], =-=[3]-=-, [6]). In the author’s view this is because the regularity lemma is a relatively “soft” component of the theory; in the infinitary framework, the “hard” components of the theory are now isolated in t... |

18 | The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of
- Kra
- 2006
(Show Context)
Citation Context ... primes contained arbitrarily long progressions; this argument was almost entirely finitary in nature, yet at the same time it relied heavily on ideas from the infinitary world of ergodic theory (see =-=[17]-=-, [15] for further discussion of this connection). In this paper we investigate a transference in the other direction, taking results from the finitary world of combinatorics (and in particular graph ... |

18 | The Gaussian primes contain arbitrarily shaped constellations
- Tao
- 2006
(Show Context)
Citation Context ... a number of applications, most notably in giving a proof not only of Szemerédi’s theorem (Theorem 2.1) but also a multidimensional version due to Furstenberg and Katznelson [9]; see e.g. [23], [12], =-=[31]-=- for further discussion of this connection, and [21] for some more applications of this theorem. A variants26 TERENCE TAO of this theorem was also used in [31] to establish that the Gaussian primes co... |

14 |
Sets of lattice points that form no squares
- Ajtai, Szemerédi
- 1974
(Show Context)
Citation Context ... a different concept of graph limit developed by Lovász and Szegedy in [18], in which the limiting object becomes a “continuous weighted graph”, or more precisely a symmetric measurable function from =-=[0,1]-=- × [0,1] to [0,1]. Such a concrete limiting object is particularly useful for computations such as counting the number of induced subgraphs of a certain shape; it also can be used to establish results... |

12 |
Note on a generalization of Roth’s theorem,Discrete and computational geometry, 825–827, Algorithms Combin. 25
- Solymosi
- 2003
(Show Context)
Citation Context ...ly finitary graph statements such as Lemma 4.1; later we shall see that one can also deduce finitary hypergraph statements in this manner. It is also well known (see [24], [5], [6], [22], [23], [12], =-=[25]-=-, [31]) that these graph and hypergraph statements can in turn be used to deduce density results such as Szemerédi’s theorem. This in turn is known by the Furstenberg correspondence principle to be eq... |

11 |
The uniformity lemma for hypergraphs, Graphs Combin
- Frankl, Rödl
- 1992
(Show Context)
Citation Context ... hypergraph regularity lemma, which has only been satisfactorily achievedsCORRESPONDENCE BETWEEN GRAPH THEORY AND PROBABILITY THEORY 25 quite recently [19], [20], [12], [30] (with preliminary work in =-=[5]-=-, [3], [6]). In the author’s view this is because the regularity lemma is a relatively “soft” component of the theory; in the infinitary framework, the “hard” components of the theory are now isolated... |

9 |
A variant of the hypergraph removal lemma, preprint
- Tao
(Show Context)
Citation Context ...a and Szemerédi [24], as well as the substantially more difficult hypergraph removal lemma of Nagle, Rödl, Schacht, and Skokan [19], [20], [22], [23] and Gowers [12] (as well as a later refinement in =-=[30]-=-). As this lemma is already strong enough to deduce Szemerédi’s theorem on arithmetic progressions (as well as a multidimensional generalisation due to Furstenberg and Katznelson [9]), we have thus pr... |

8 | Regular partitions of hypergraphs, preprint - Rödl, Schacht |

6 | Progressions arithmétiques dans les nombres premiers (d’après B. Green et T - Host |

6 | Density theorems and extremal hypergraph problems, manuscript
- Rödl, Schacht, et al.
(Show Context)
Citation Context ...proof not only of Szemerédi’s theorem (Theorem 2.1) but also a multidimensional version due to Furstenberg and Katznelson [9]; see e.g. [23], [12], [31] for further discussion of this connection, and =-=[21]-=- for some more applications of this theorem. A variants26 TERENCE TAO of this theorem was also used in [31] to establish that the Gaussian primes contain arbitrarily shaped constellations; we shall di... |

6 |
Szemeredi’s regularity lemma revisited, preprint
- Tao
- 2005
(Show Context)
Citation Context ...c or “L logLbased” method, manipulating (conditional) entropies and mutual informations instead, relying on basic information-theoretic facts such as subadditivity of entropy and Bayes’ identity. See =-=[28]-=- for a comparison between the two techniques. However we shall stick purely to the L 2 approach here; the information-theoretic approach is perhaps conceptually clearer, but requires one to discuss en... |

2 |
An ergodic transference theorem, unpublished
- Tao
(Show Context)
Citation Context ...ctness and sequential compactness). Observe that both the Heine-Borel theorem and the Kolmogorov extension theorem are completely constructive, and so this lemma does not use the axiom of choice. See =-=[32]-=- for further discussion. A.17. Approximation lemmas. We will frequently need to approximate a random variable or event in a complicated factor by linear, polynomial, or boolean combinations of random ... |

2 |
Herbrand’s theorem and proof theory
- Girard
- 1982
(Show Context)
Citation Context ...on due to Furstenberg and Katznelson [9]), we have thus presented yet another proof of Szemerédi’s theorem here. These lemmas have some further applications; for theorem via topological dynamics, see =-=[11]-=-. However such methods do not seem to shed much light on the connection between the infinitary proofs and the existing finitary proofs in the literature. 2 This is actually not all that surprising, gi... |

1 |
Herbrand’s theorem and proof theory, Proceedings of the Herbrand symposium
- Girard
- 1981
(Show Context)
Citation Context ...on due to Furstenberg and Katznelson [9]), we have thus presented yet another proof of Szemerédi’s theorem here. These lemmas have some further applications; for theorem via topological dynamics, see =-=[11]-=-. However such methods do not seem to shed much light on the connection between the infinitary proofs and the existing finitary proofs in the literature. 2 This is actually not all that surprising, gi... |

1 |
Limits of dense graph sequences, preprint
- Lovász, Szegedy
(Show Context)
Citation Context ...bility theory has already proven to have a major role to play in graph theory. 3 This is related to, but slightly different from, a different concept of graph limit developed by Lovász and Szegedy in =-=[18]-=-, in which the limiting object becomes a “continuous weighted graph”, or more precisely a symmetric measurable function from [0,1] × [0,1] to [0,1]. Such a concrete limiting object is particularly use... |