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31
Szemerédi's Regularity Lemma and Its Applications in Graph Theory
, 1996
"... Szemer'edi'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 q ..."
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Cited by 209 (3 self)
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Szemer'edi'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. Contents Preface 1. Introduction 2. How to apply the Regularity Lemma 3. Early applications 4. Building large subgraphs 5. Embedding trees 6. Bounded degree spanning subgraphs 7. Weakening the Regularity Lemma 8. Strengthening the Regularity Lemma 9. Algorithmic questions 10. Regularity and randomness Preface Szemer'edi's Regularity Lemma [121] is one of the most powerful tools of (extremal) graph theory. It was invented as an auxiliary lemma in the proof of the famous conjectu...
Turán Numbers of Bipartite Graphs and Related RamseyType Questions
, 2003
"... For a graph H and an integer n, theTurán number ex(n, H) isthemaximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is rdegenerate if every one of its subgraphs contains a vertex of degree at most r. Weprove that, for any fixed bipartite graph H in which all ..."
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Cited by 31 (21 self)
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For a graph H and an integer n, theTurán number ex(n, H) isthemaximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is rdegenerate if every one of its subgraphs contains a vertex of degree at most r. Weprove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, ex(n, H) �O(n 2−1/r). This is tight for all values of r and can also be derived from an earlier result of Füredi. We also show that there is an absolute positive constant c such that, for every fixed bipartite rdegenerate graph H, ex(n, H) �O(n 1−c/r). This is motivated by aconjectureofErdős that asserts that, for every such H, ex(n, H) �O(n 1−1/r). For two graphs G and H, the Ramsey number r(G, H) istheminimum number n such that, in any colouring of the edges of the complete graph on n vertices by red and blue, there is either aredcopy of G or a blue copy of H. Erdős conjectured that there is an absolute constant c such that, for any graph G with m edges, r(G, G) �2 c √ m.Hereweprove this conjecture for bipartite graphs G, andprove that for general graphs G with m edges,
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 13 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Additive approximation for edgedeletion problems
 Proc. of FOCS 2005
, 2005
"... A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edgedeletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G ..."
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Cited by 11 (8 self)
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A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edgedeletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E ′ P (G). The first result of this paper states that the edgedeletion problem can be efficiently approximated for any monotone property. • For any fixed ɛ> 0 and any monotone property P, there is a deterministic algorithm, which given a graph G = (V, E) of size n, approximates E ′ P (G) in linear time O(V  + E) to within an additive error of ɛn2. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E ′ P. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist. 1. If there is a bipartite graph that does not satisfy P, then there is a δ> 0 for which it is
Stability theorems for cancellative hypergraphs
 J. Combin. Theory Ser. B
"... A cancellative hypergraph has no three edges A, B, C with A∆B ⊂ C. We give a new short proof of an old result of Bollobás, which states that the maximum size of a cancellative triple system is achieved by the balanced complete tripartite 3graph. One of the two forbidden subhypergraphs in a cancella ..."
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Cited by 11 (6 self)
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A cancellative hypergraph has no three edges A, B, C with A∆B ⊂ C. We give a new short proof of an old result of Bollobás, which states that the maximum size of a cancellative triple system is achieved by the balanced complete tripartite 3graph. One of the two forbidden subhypergraphs in a cancellative 3graph is F5 = {abc, abd, cde}. For n ≥ 33 we show that the maximum number of triples on n vertices containing no copy of F5 is also achieved by the balanced complete tripartite 3graph. This strengthens a theorem of Frankl and Füredi, who proved it for n ≥ 3000. For both extremal results, we show that a 3graph with almost as many edges as the extremal example is approximately tripartite. These stability theorems are analogous to the Simonovits stability theorem for graphs. 1
SetSystems with Restricted Multiple Intersections and Explicit Ramsey Hypergraphs
, 2001
"... We give a generalization for the DezaFranklSinghi Theorem in case of multiple intersections. More exactly, we prove, that if H is a setsystem, which satisfies that for some k, the kwise intersections occupy only ` residueclasses modulo a p prime, while the sizes of the members of H are not in t ..."
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Cited by 8 (1 self)
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We give a generalization for the DezaFranklSinghi Theorem in case of multiple intersections. More exactly, we prove, that if H is a setsystem, which satisfies that for some k, the kwise intersections occupy only ` residueclasses modulo a p prime, while the sizes of the members of H are not in these residue classes, then the size of H is at most (k \Gamma 1) ` X i=0 / n i ! This result considerably strengthens an upper bound of Furedi (1983), and gives partial answer to a question of T. S'os (1976). As an application, we give explicit constructions for coloring the ksubsets of an n element set with t colors, such that no monochromatic complete hypergraph on exp(c k (log n log log n) 1=t ) vertices exists. By our best knowledge, this is the first explicit construction of a Ramseyhypergraph.
Geometric Graphs With No SelfIntersecting Path Of Length Three
"... Let G be a geometric graph with n vertices, i.e., a graph drawn in the plane with straightline edges. It is shown that if G has no selfintersecting path of length 3, then its number of edges is O(n log n). This result is asymptotically tight. Analogous questions for curvilinear drawings and for ..."
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Cited by 8 (5 self)
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Let G be a geometric graph with n vertices, i.e., a graph drawn in the plane with straightline edges. It is shown that if G has no selfintersecting path of length 3, then its number of edges is O(n log n). This result is asymptotically tight. Analogous questions for curvilinear drawings and for longer paths are also considered.