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133
On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27
, 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 251 (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 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.
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 53 (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.
Problems and results in combinatorial analysis
 COMBINATORICS (PROC. SYMP. PURE MATH
, 1971
"... This review of some solved and unsolved problems in combinatorial analysis will be highly subjective. i will only discuss problems which I either worked on or at least thought about. The disadvantages of such an approach are obvious, but the disadvantages are perhaps counterbalanced by the fact that ..."
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Cited by 40 (0 self)
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This review of some solved and unsolved problems in combinatorial analysis will be highly subjective. i will only discuss problems which I either worked on or at least thought about. The disadvantages of such an approach are obvious, but the disadvantages are perhaps counterbalanced by the fact that I certainly know more about these problems than about others (which perhaps are more important). i will mainly discuss finite combinatorial problems. I cannot claim completeness in any way but will try to refer to the literature in some cases; even so many things will be omitted. ISO will denote the cardinal number of S; c, cl, c2,... will denote absolute constants not necessarily the same at each occurrence. I. I will start with some problems dealing with subsets of a set. Let IS I =n. A well known theorem of Sperner [57] states that if A i a S, 15 i 5 m, is such that no A, contains any other, then max m=(aA). The theorem of Sperner has many applications in number theory; as far as I know these were first noticed by Behrend [2] and myself [8]. I asked 30 years ago several further extremal problems about subsets which also have number theoretic consequences. Let At a S, 15 i 5mi, assume that there are no three distinct A's so that Ai V A! = A,. I conjectured that
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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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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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.
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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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.
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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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.
Setpolynomials and polynomial extension of the HalesJewett theorem Ann
 of Math
, 1999
"... An abstract, HalesJewett type extension of the polynomial van der Waerden Theorem [BL] is established: Theorem. Let r,d,q ∈ N. There exists N ∈ N such that for any rcoloring of the set of subsets of V = {1,...,N} d × {1,...,q} there exist a set a ⊂ V and a nonempty set γ ⊆ {1,...,N} such that a ∩ ..."
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An abstract, HalesJewett type extension of the polynomial van der Waerden Theorem [BL] is established: Theorem. Let r,d,q ∈ N. There exists N ∈ N such that for any rcoloring of the set of subsets of V = {1,...,N} d × {1,...,q} there exist a set a ⊂ V and a nonempty set γ ⊆ {1,...,N} such that a ∩ (γ d × {1,...,q}) = ∅, and the subsets a, a ∪ (γ d × {1}), a ∪ (γ d × {2}),..., a ∪ (γ d × {q}) are all of the same color. This “polynomial ” HalesJewett theorem contains refinements of many combinatorial facts as special cases. The proof is achieved by introducing and developing the apparatus of setpolynomials (polynomials whose coefficients are finite sets) and applying the methods of topological dynamics. 0.1. The celebrated van der Waerden Theorem ([W]) states that if the