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44
Szemerédi's Regularity Lemma and Its Applications in Graph Theory
, 1996
"... Szemer'edi's Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by randomlooking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. Recently q ..."
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Cited by 209 (3 self)
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Szemer'edi's Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by randomlooking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. Recently quite a few new results were obtained by using the Regularity Lemma, and also some new variants and generalizations appeared. In this survey we describe some typical applications and some generalizations. Contents Preface 1. Introduction 2. How to apply the Regularity Lemma 3. Early applications 4. Building large subgraphs 5. Embedding trees 6. Bounded degree spanning subgraphs 7. Weakening the Regularity Lemma 8. Strengthening the Regularity Lemma 9. Algorithmic questions 10. Regularity and randomness Preface Szemer'edi's Regularity Lemma [121] is one of the most powerful tools of (extremal) graph theory. It was invented as an auxiliary lemma in the proof of the famous conjectu...
The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
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Cited by 150 (26 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Integer sets containing no arithmetic progressions
 J. London Math. Soc
, 1987
"... lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of length k. This is the famous theorem of van der Waerden [10], dating from 1927. The proof of this uses mu ..."
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Cited by 49 (0 self)
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lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of length k. This is the famous theorem of van der Waerden [10], dating from 1927. The proof of this uses multiple nested inductions, which result
A variant of the hypergraph removal lemma
, 2006
"... Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and FurstenbergKatznelson [7] concerning onedimensional and multidimensional arithmetic progressions respecti ..."
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Cited by 47 (4 self)
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Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and FurstenbergKatznelson [7] concerning onedimensional and multidimensional arithmetic progressions respectively. In this paper we shall give a selfcontained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [29] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes. 1.
A quantitative ergodic theory proof of Szemerédi’s theorem
, 2004
"... A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions of length k. For general k there are essentially four known proofs of this fact; Szemerédi’s original combinato ..."
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Cited by 34 (14 self)
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A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions of length k. For general k there are essentially four known proofs of this fact; Szemerédi’s original combinatorial proof using the Szemerédi regularity lemma and van der Waerden’s theorem, Furstenberg’s proof using ergodic theory, Gowers’ proof using Fourier analysis and the inverse theory of additive combinatorics, and Gowers’ more recent proof using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring in particular the use of transfinite induction (and thus the axiom of choice), decomposing a general ergodic system as the weakly mixing extension of a transfinite tower of compact extensions. Here we present a quantitative, selfcontained version of this ergodic theory proof, and which is “elementary ” in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.
A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma, preprint
"... 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 ..."
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Cited by 21 (5 self)
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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 NagleSchachtRödlSkokan 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.
The Gaussian primes contain arbitrarily shaped constellations
 J. dAnalyse Mathematique
"... Abstract. We show that the Gaussian primes P[i] ⊆ Z[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers v0,..., vk−1, we show that there are infinitely many sets {a+rv0,..., a+rvk−1}, with a ∈ Z[i] and r ∈ Z\{0}, all of ..."
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Cited by 18 (10 self)
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Abstract. We show that the Gaussian primes P[i] ⊆ Z[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers v0,..., vk−1, we show that there are infinitely many sets {a+rv0,..., a+rvk−1}, with a ∈ Z[i] and r ∈ Z\{0}, all of whose elements are Gaussian primes. The proof is modeled on that in [9] and requires three ingredients. The first is a hypergraph removal lemma of Gowers and RödlSkokan, or more precisely a slight strengthening of this lemma which can be found in [22]; this hypergraph removal lemma can be thought of as a generalization of the SzemerédiFurstenbergKatznelson theorem concerning multidimensional arithmetic progressions. The second ingredient is the transference argument from [9], which allows one to extend this hypergraph removal lemma to a relative version, weighted by a pseudorandom measure. The third ingredient is a GoldstonYıldırım type analysis for the Gaussian integers, similar to that in [9], which yields a pseudorandom measure which is concentrated on Gaussian “almost primes”. 1.
Szemerédi’s regularity lemma revisited
 Contrib. Discrete Math
"... 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 inf ..."
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Cited by 14 (3 self)
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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.