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75
The ergodic and combinatorial approaches to Szemerédi’s theorem
, 2006
"... A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graphtheoretical) approach of Szem ..."
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A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graphtheoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourieranalytic approach of Gowers, and the hypergraph approach of NagleRödlSchachtSkokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different.
A proof of Green’s conjecture regarding the removal properties of sets of linear equations
"... A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) solutions of Mx = b can be turned into a set S ′ containing no solution of Mx = b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homo ..."
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A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) solutions of Mx = b can be turned into a set S ′ containing no solution of Mx = b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homogenous linear equation always has the removal property, and conjectured that every set of homogenous linear equations has the removal property. In this paper we confirm Green’s conjecture by showing that every set of linear equations (even nonhomogenous) has the removal property. We also discuss some applications of our result in theoretical computer science, and in particular, use it to resolve a conjecture of Bhattacharyya, Chen, Sudan and Xie [7] related to algorithms for testing properties of boolean functions.
DECOMPOSITIONS, APPROXIMATE STRUCTURE, TRANSFERENCE, AND THE HAHNBANACH THEOREM
, 2008
"... We discuss three major classes of theorems in additive and extremal combinatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the finitedimensional HahnBanach theorem can be used to give short and transparent proofs of many results of the ..."
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We discuss three major classes of theorems in additive and extremal combinatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the finitedimensional HahnBanach theorem can be used to give short and transparent proofs of many results of these kinds. Amongst the applications of this method is a much shorter proof of one of the major steps in the proof of Green and Tao that the primes contain arbitrarily long arithmetic progressions. In order to explain the role of this step, we include a brief description of the rest of their argument. A similar proof has been discovered independently by Reingold, Trevisan, Tulsiani and Vadhan [RTTV].
Stability results for random discrete structures, Random Structures Algorithms
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Open problems in additive combinatorics
"... A brief historical introduction to the subject of additive combinatorics and a list of challenging open problems, most of which are contributed by the leading experts in the area, are presented. ..."
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Cited by 13 (0 self)
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A brief historical introduction to the subject of additive combinatorics and a list of challenging open problems, most of which are contributed by the leading experts in the area, are presented.
Extremal results in sparse pseudorandom graphs
 ADV. MATH. 256 (2014), 206–290
, 2014
"... Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extendin ..."
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Cited by 13 (8 self)
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Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi’s regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a wellknown open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rödl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several wellknown combinatorial theorems, including the removal lemmas for graphs and groups, the ErdősStoneSimonovits theorem and Ramsey’s
An algorithmic version of the hypergraph regularity method
 PROCEEDINGS OF THE IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2005
"... Extending the Szemerédi Regularity Lemma for graphs, P. Frankl and V. Rödl [14] established a 3graph Regularity Lemma guaranteeing that all large triple systems Gn admit bounded partitions of their edge sets, most classes of which consist of regularly distributed triples. Many applications of t ..."
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Cited by 11 (7 self)
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Extending the Szemerédi Regularity Lemma for graphs, P. Frankl and V. Rödl [14] established a 3graph Regularity Lemma guaranteeing that all large triple systems Gn admit bounded partitions of their edge sets, most classes of which consist of regularly distributed triples. Many applications of this lemma require a companion Counting Lemma [30], allowing one to find and enumerate subhypergraphs of a given isomorphism type in a “dense and regular” environment created by the 3graph Regularity Lemma. Combined applications of these lemmas are known as the 3graph Regularity Method. In this paper, we provide an algorithmic version of the 3graph Regularity Lemma which, as we show, is compatible with a Counting Lemma. We also discuss some applications.
Property testing in hypergraphs and the removal lemma (Extended Abstract)
, 2006
"... Property testers are efficient, randomized algorithms which recognize if an input graph (or other combinatorial structure) satisfies a given property or if it is “far” from exhibiting it. Generalizing several earlier results, Alon and Shapira showed that hereditary graph properties are testable (wit ..."
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Cited by 11 (0 self)
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Property testers are efficient, randomized algorithms which recognize if an input graph (or other combinatorial structure) satisfies a given property or if it is “far” from exhibiting it. Generalizing several earlier results, Alon and Shapira showed that hereditary graph properties are testable (with onesided error). In this paper we prove the analogous result for hypergraphs. This result is an immediate consequence of a (hyper)graph theoretic statement, which is an extension of the socalled removal lemma. The proof of this generalization relies on the regularity method for hypergraphs.
Graph removal lemmas
 SURVEYS IN COMBINATORICS
, 2013
"... The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and com ..."
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Cited by 9 (3 self)
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The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.