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166
On sets of integers containing no k elements in arithmetic progression
, 1975
"... integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequen ..."
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Cited by 286 (1 self)
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integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequence bn9 with bn> an9 which contains no arithmetic progression of three terms, but which intersects every infinite arithmetic progression. The finite form of van der Waerden's theorem goes as follows: For each positive integer n9 there exists a least integer f{n) with the property that if the integers from 1 to /(/?) are arbitrarily partitioned into two classes, then at least one class contains an arithmetic progression of « terms. (For a short proof, see the note of Graham and Rothschild [5].) However, the best upper bound on f{n) known at present is extremely poor. The best lower bound known, due to Berlekamp [3], asserts that/(«) < nln9 for n prime, which improves previous results of Erdös, Rado and W. Schmidt. More than 40 years ago, Erdös and Turân [4] considered the quantity rk{n)9 defined to be the greatest integer / for which there is a sequence of integers 0 < a \ < a2 < •• • < a; ^ n which does not contain an arithmetic progression of k terms. They were led to the investigation of rk{n) by several things. First of all the problem of estimating rk{n) is clearly interesting in itself. Secondly, rk{n) < n/2 would imply f{k) < 77, i.e., they hoped to improve the poor upper bound on f{k) by investigating rk{n). Finally, an old question in number theory asks if there are arbitrarily long arithmetic progressions of prime numbers. From rk{n) < %{rì) this would follow immediately. The hope was that this problem on primes could be attacked not by
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 224 (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 194 (29 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
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 57 (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.
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 53 (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
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 ..."
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Cited by 52 (15 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.
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. E ..."
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Cited by 48 (3 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.