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17
A simple regularization of hypergraphs
"... Abstract. We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well a ..."
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Abstract. We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well as its multidimensional extension by FurstenbergKatznelson (1978). 1.
Removal lemma for infinitelymany forbidden hypergraphs and property testing
, 2008
"... We prove a removal lemma for infinitelymany forbidden hypergraphs. It affirmatively settles a question on property testing raised by Alon and Shapira (2005) [2, 3]. All monotone hypergraph properties and all hereditary partite hypergraph properties are testable. Our proof constructs a constanttim ..."
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We prove a removal lemma for infinitelymany forbidden hypergraphs. It affirmatively settles a question on property testing raised by Alon and Shapira (2005) [2, 3]. All monotone hypergraph properties and all hereditary partite hypergraph properties are testable. Our proof constructs a constanttime probabilistic algorithm to edit a small number of edges. It also gives a quantitative bound in terms of a coloring number of the property. It is based on a new hypergraph regularity lemma [14].
Additive Combinatorics and Theoretical Computer Science
, 2009
"... Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties ..."
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Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties and patterns that can be expressed via additions and multiplications. In the past ten years, additive and arithmetic combinatorics have been extremely successful areas of mathematics, featuring a convergence of techniques from graph theory, analysis and ergodic theory. They have helped prove longstanding open questions in additive number theory, and they offer much promise of future progress. Techniques from additive and arithmetic combinatorics have found several applications in computer science too, to property testing, pseudorandomness, PCP constructions, lower bounds, and extractor constructions. Typically, whenever a technique from additive or arithmetic combinatorics becomes understood by computer scientists, it finds some application. Considering that there is still a lot of additive and arithmetic combinatorics that computer scientists do not understand (and, the field being very active, even more will be developed in the near future), there seems to be much potential for future connections and applications.
What is good mathematics
, 2007
"... Abstract. Some personal thoughts and opinions on what “good quality mathematics” is, and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem is presented. 1. The many aspects of mathematical quality We all agree that mathematicians should strive t ..."
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Abstract. Some personal thoughts and opinions on what “good quality mathematics” is, and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem is presented. 1. The many aspects of mathematical quality We all agree that mathematicians should strive to produce good mathematics. But how does one define “good mathematics”, and should one even dare to try at all? Let us first consider the former question. Almost immediately one realises that there are many different types of mathematics which could be designated “good”. For instance, “good mathematics ” could refer (in no particular order) to (i) Good mathematical problemsolving (e.g. a major breakthrough on an important mathematical problem); (ii) Good mathematical technique (e.g. a masterful use of existing methods, or the development of new tools); (iii) Good mathematical theory (e.g. a conceptual framework or choice of notation which systematically unifies and generalises an existing body of results);
COMBINATORIAL INDEPENDENCE IN MEASURABLE DYNAMICS
"... Abstract. We develop a finescale local analysis of measure entropy and measure sequence entropy based on combinatorial independence. The concepts of measure IEtuples and measure INtuples are introduced and studied in analogy with their counterparts in topological dynamics. Local characterizations ..."
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Abstract. We develop a finescale local analysis of measure entropy and measure sequence entropy based on combinatorial independence. The concepts of measure IEtuples and measure INtuples are introduced and studied in analogy with their counterparts in topological dynamics. Local characterizations of the Pinsker von Neumann algebra and its sequence entropy analogue are given in terms of combinatorial independence, ℓ1 geometry, and Voiculescu’s completely positive approximation entropy. Among the novel features of our local study is the treatment of general discrete acting groups, with the structural assumption of amenability in the case of entropy. 1.
ON NOWHERE DENSE GRAPHS
"... A set A of vertices of a graph G is called dscattered in G if no two dneighborhoods of (distinct) vertices of A intersect. In other words, A is dscattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and rec ..."
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A set A of vertices of a graph G is called dscattered in G if no two dneighborhoods of (distinct) vertices of A intersect. In other words, A is dscattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed families). This in turn led to the notions of wide, semiwide and quasiwide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativised) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) classes of nowhere dense graphs are quasi wide. This not only strictly generalizes the previous results and solves several open problems but it also provides new proofs. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of widetype classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context and we are able to prove a trichotomy result which separates classes of graphs which are dense (somewhere dense), nowhere dense and finite. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a duality.
Additive Combinatorics with a view towards Computer Science and Cryptography An
, 2011
"... Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is ..."
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Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of additive structures in sets equipped with a group structure – we may have other structure that interacts with this group structure. This newly emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply crossfertilize each other, holding great promise for all of them! There is a considerable number of incredible problems, results, and novel applications in this thriving area. In this exposition, we attempt to provide an illuminating overview of some conspicuous breakthroughs in this captivating field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.