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53
Szemerédi's Regularity Lemma and Its Applications in Graph Theory
, 1996
"... Szemerédi'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. Recent ..."
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Cited by 217 (3 self)
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Szemerédi'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.
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 169 (27 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
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 combin ..."
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Cited by 37 (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.
The GreenTao Theorem on arithmetic progressions in the primes: an ergodic point of view
, 2005
"... A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an a ..."
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Cited by 28 (2 self)
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A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.
The Gaussian primes contain arbitrarily shaped constellations
 J. Anal. Math
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A new proof of the density HalesJewett theorem
, 2009
"... The Hales–Jewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1,..., k} n contains a combinatorial line. This result is a generalization of van der Waerden’s theorem, and it is one of the fundamental results of Ramsey theory. The ..."
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Cited by 18 (1 self)
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The Hales–Jewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1,..., k} n contains a combinatorial line. This result is a generalization of van der Waerden’s theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van der Waerden has a famous density version, conjectured by Erdős and Turán in 1936, proved by Szemerédi in 1975 and given a different proof by Furstenberg in 1977. The Hales–Jewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemerédi’s theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson, and the first to provide a quantitative bound on how large n needs to be. In particular, we show that a subset of [3] n of density δ contains a combinatorial line if n ≥ 2 ⇈ O(1/δ 3). Our proof is surprisingly simple: indeed, it gives what is probably the simplest known proof of Szemerédi’s theorem.
Tight Lower Bounds for the Size of EpsilonNets
"... According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O () ..."
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Cited by 12 (1 self)
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According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O ()
Zerosum problems in finite abelian groups and affine caps
 QUARTERLY JOURNAL OF MATHEMATICS
, 2007
"... For a finite abelian group G, let s(G) denote the smallest integer l such that every sequence S over G of length S  ≥l has a zerosum subsequence of length exp(G). We derive new upper and lower bounds for s(G), and all our bounds are sharp for special types of groups. The results are not restricte ..."
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Cited by 12 (3 self)
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For a finite abelian group G, let s(G) denote the smallest integer l such that every sequence S over G of length S  ≥l has a zerosum subsequence of length exp(G). We derive new upper and lower bounds for s(G), and all our bounds are sharp for special types of groups. The results are not restricted to groups G of the form G = Cr n, but they respect the structure of the group. In particular, we show s(C4 n) ≥ 20n − 19 for all odd n, which is sharp if n is a power of 3. Moreover, we investigate the relationship between extremal sequences and maximal caps in finite geometry.
New bounds for Szemerédi’s theorem, I: Progressions of length 4 in finite field geometries
 Proc. Lond. Math. Soc
"... Abstract. Let k � 3 be an integer, and let G be a finite abelian group with G  = N, where (N, (k − 1)!) = 1. We write rk(G) for the largest cardinality A  of a set A ⊆ G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts th ..."
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Cited by 11 (5 self)
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Abstract. Let k � 3 be an integer, and let G be a finite abelian group with G  = N, where (N, (k − 1)!) = 1. We write rk(G) for the largest cardinality A  of a set A ⊆ G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts that rk(Z/NZ) = ok(N). It is known, in fact, that the estimate rk(G) = ok(N) holds for all G. There have been many papers concerning the issue of finding quantitative bounds for rk(G). A result of Bourgain states that r3(G) ≪ N(log log N / logN) 1/2 for all G. In this paper we obtain a similar bound for r4(G) in the particular case G = F n, where F is a fixed finite field with char(F) ̸ = 2, 3 (for example, F = F5). We prove that r4(G) ≪F N(log N) −c for some absolute constant c> 0. In future papers we will treat general abelian groups G, eventually obtaining a comparable result for arbitrary G. 1.