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101
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 239 (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.
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 77 (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.
Monotonicity testing over general poset domains (Extended Abstract)
 STOC'02
, 2002
"... The field of property testing studies algorithms that distinguish, using a small number of queries, between inputs which satisfy a given property, and those that are ‘far’ from satisfying the property. Testing properties that are defined in terms of monotonicity has been extensively investigated, pr ..."
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Cited by 58 (26 self)
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The field of property testing studies algorithms that distinguish, using a small number of queries, between inputs which satisfy a given property, and those that are ‘far’ from satisfying the property. Testing properties that are defined in terms of monotonicity has been extensively investigated, primarily in the context of the monotonicity of a sequence of integers, or the monotonicity of a function over the £dimensional hypercube ¤¥¦§§ § ¦¨©�. These works resulted in monotonicity testers whose query complexity is at most polylogarithmic in the size of the domain. We show that in its most general setting, testing that Boolean functions are close to monotone is equivalent, with respect to the number of required queries, to several other testing problems in logic and graph theory. These problems include: testing that a Boolean assignment of variables is close to an assignment that satisfies a specific �CNF formula, testing that a set of vertices is close to one that is a vertex cover of a specific graph, and testing that a set of vertices is close to a clique. We then investigate the query complexity of monotonicity testing of both Boolean and integer functions over general partial orders. We give algorithms and lower bounds for the general problem, as well as for some interesting special cases. In proving a general lower bound, we construct graphs with combinatorial properties that may be of independent interest.
Satisfiability Allows No Nontrivial Sparsification Unless The PolynomialTime Hierarchy Collapses
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 38 (2010)
, 2010
"... Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small ..."
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Cited by 54 (2 self)
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Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer d ≥ 3 and positive real ǫ we show that if satisfiability for nvariable dCNF formulas has a protocol of cost O(n d−ǫ) then coNP is in NP/poly, which implies that the polynomialtime hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for ǫ = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NPcomplete problems. For the vertex cover problem on nvertex duniform hypergraphs, the above statement holds for any integer d ≥ 2. The case d = 2 implies that no NPhard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k 2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and boundeddegree deletion.
Arithmetic progressions of length three in subsets of a random set
 ACTA ARITH
, 1996
"... For integers 1 ≤ M ≤ n, let R(n, M) denote the uniform probability space which consists of all the Melement subsets of [n] = {0, 1,..., n − 1}. It is shown that for every α> 0 there exists a constant C such that if M = M(n) ≥ C √ n then, with probability tending to 1 as n → ∞, the random set ..."
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Cited by 51 (9 self)
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For integers 1 ≤ M ≤ n, let R(n, M) denote the uniform probability space which consists of all the Melement subsets of [n] = {0, 1,..., n − 1}. It is shown that for every α> 0 there exists a constant C such that if M = M(n) ≥ C √ n then, with probability tending to 1 as n → ∞, the random set R ∈ R(n, M) has the property that any subset of R with at least αR  elements contains a 3term arithmetic progression. In particular, this result implies that for every α> 0 there exist ‘sparse ’ sets S ⊆ [n] with the property that every subset of S with at least αS  elements contains an arithmetic progression of length three.
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 46 (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.
Applications of the Regularity Lemma for UNIFORM HYPERGRAPHS
, 2004
"... In this note we discuss several combinatorial problems that can be addressed by the Regularity Method for hypergraphs. Based on recent results of Nagle, Schacht and the authors, we give here solutions to these problems. In particular, we prove the following: Let K (k) t be the complete kuniform h ..."
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Cited by 41 (5 self)
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In this note we discuss several combinatorial problems that can be addressed by the Regularity Method for hypergraphs. Based on recent results of Nagle, Schacht and the authors, we give here solutions to these problems. In particular, we prove the following: Let K (k) t be the complete kuniform hypergraph on t vertices and suppose an nvertex kuniform hypergraph H contains only o(n t) copies of K (k) t. Then one can delete o(n k) edges of H to make it K (k) tfree. Similar results were recently obtained by W. T. Gowers.
ON THE COMBINATORIAL PROBLEMS WHICH I WOULD MOST LIKE TO SEE SOLVED
, 1979
"... I was asked to write a paper about the major unsolved problems in combinatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no re ..."
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Cited by 37 (0 self)
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I was asked to write a paper about the major unsolved problems in combinatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no real competence but also topics about which I never seriously thought, e.g. algorithmic combinatorics, coding theory and matroid theory. There is no doubt that the proof of the conjecture that several simply stated problems have no good algorithm is fundamental and may have important consequences for many other branches of mathematics, but unfortunately I have no real feeling for these questions and I feel I should leave the subject to those who are more competent. I just heard that Khachiyan [59], has a polynomial algorithm for linear programming. (See also [50].) This is considered a sensational result and during my last stay in the U.S. many of my friends were greatly impressed by it.