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357
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 257 (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 algorithmic aspects of the Regularity Lemma
 J. Algorithms
, 1994
"... The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that ..."
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Cited by 108 (30 self)
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The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that deciding if a given partition of an input graph satisfies the properties guaranteed by the lemma is coNPcomplete. However, we also prove that despite this difficulty the lemma can be made constructive; we show how to obtain, for any input graph, a partition with the properties guaranteed by the lemma, efficiently. The desired partition, for an nvertex graph, can be found in time O(M(n)), where M(n) = O(n 2.376) is the time needed to multiply two n by n matrices with 0, 1entries over the integers. The algorithm can be parallelized and implemented in NC 1. Besides the curious phenomenon of exhibiting a natural problem in which the search for a solution is easy whereas the decision if a given instance is a solution is difficult (if P and NP differ), our constructive version of the Regularity Lemma supplies efficient sequential and parallel algorithms for many problems, some of which are naturally motivated by the study of various graph embedding and graph coloring problems.
The counting lemma for regular kuniform hypergraphs
, 2004
"... Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of ..."
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Cited by 105 (14 self)
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Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of density d for ℓ 1 ≤ i < j ≤ ℓ, then G contains (1 ± fℓ(ε))d
Regularity lemma for kuniform hypergraphs, Random Structures and Algorithms
, 2005
"... Szemerédi’s Regularity Lemma proved to be a very powerful tool in extremal graph theory with a large number of applications. Chung [Regularity lemmas for hypergraphs and quasirandomness, Random ..."
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Cited by 90 (8 self)
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Szemerédi’s Regularity Lemma proved to be a very powerful tool in extremal graph theory with a large number of applications. Chung [Regularity lemmas for hypergraphs and quasirandomness, Random
Norm convergence of multiple ergodic averages for commuting transformations
, 2007
"... Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established fo ..."
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Cited by 81 (4 self)
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Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [3] (with the l = 3 case of this result established earlier in [29]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l = 2 case of our arguments are a finitary analogue of those in [2].
Linear equations in primes
 ANNALS OF MATHEMATICS
, 2006
"... Consider a system Ψ of nonconstant affinelinear forms ψ1,..., ψt: Z d → Z, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [−N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N → ∞, for th ..."
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Cited by 79 (3 self)
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Consider a system Ψ of nonconstant affinelinear forms ψ1,..., ψt: Z d → Z, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [−N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N → ∞, for the number of integer points n ∈ Z d ∩ K for which the integers ψ1(n),..., ψt(n) are simultaneously prime. This implies many other wellknown conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affinelinear forms ψ1,..., ψt are affinely related; this excludes the important “binary ” cases such as the twin prime or Goldbach conjectures, but does allow one to count “nondegenerate ” configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowersnorm conjecture (GI(s)) and the Möbius and nilsequences conjecture (MN(s)), where s ∈ {1, 2,...} is
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 76 (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 75 (7 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.