Results 1  10
of
122
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.
A NEW PROOF OF SZEMERÉDI’S THEOREM FOR ARITHMETIC PROGRESSIONS OF LENGTH FOUR
 GAFA, GEOMETRIC AND FUNCTIONAL ANALYSIS
, 1998
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Szemerédi’s regularity lemma for sparse graphs
 Foundations of Computational Mathematics
, 1997
"... A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This farreaching result has proved to play a central r^ole in many areas of combinatorics, both `pure ' and `algorithmic. ' The quest for ..."
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Cited by 56 (20 self)
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A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This farreaching result has proved to play a central r^ole in many areas of combinatorics, both `pure ' and `algorithmic. ' The quest for an equally powerful variant of this lemma for sparse graphs has not yet been successful, but some progress has been achieved recently. The aim of this note is to report on the successes so far.
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 POLYNOMIAL BOUND IN FREIMAN’S THEOREM
 DUKE MATHEMATICAL JOURNAL VOL. 113, NO. 3
, 2002
"... In this paper the following improvement on Freiman’s theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A ⊂ Z be a finite set such that A + A  < αA. Then A is contained in a proper ddimensional progression P, where d ≤ [α − 1] and log(P/A) < Cα 2 (log α) 3. Earlier ..."
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Cited by 44 (2 self)
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In this paper the following improvement on Freiman’s theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A ⊂ Z be a finite set such that A + A  < αA. Then A is contained in a proper ddimensional progression P, where d ≤ [α − 1] and log(P/A) < Cα 2 (log α) 3. Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa’s method, which we improve in several places, as well as Y. Bilu’s proof of Freiman’s theorem.