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99
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 ..."
Abstract

Cited by 201 (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...
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 33 (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.
Supersaturated graphs and hypergraphs
 Combinatorica
, 1983
"... We shall consider graphs (hypergraphs) without loops and multiple edges. Let Ybe a family of so called prohibited graphs and ex (n, Y) denote the maximum number of edges (hyperedges) a graph (hypergraph) on n vertices can have without containing subgraphs from Y A graph (hypergraph) will be called s ..."
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Cited by 28 (0 self)
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We shall consider graphs (hypergraphs) without loops and multiple edges. Let Ybe a family of so called prohibited graphs and ex (n, Y) denote the maximum number of edges (hyperedges) a graph (hypergraph) on n vertices can have without containing subgraphs from Y A graph (hypergraph) will be called supersaturated if it has more edges than ex (n, Y). If G has n vertices and ex (n, Y)=k edges (hyperedges), then it always contains prohibited subgraphs. The basic question investigated here is: At least how many copies of L E Y must occur in a graph G " on n vertices with ex (n, Y)+k edges (hy peredges)? Notation. In this paper we shall consider only graphs and hypergraphs without loops and multiple edges, and all hypergraphs will be uniform. If G is a graph or hypergraph, e(G), v(G) and y(G) will denote the number of edges, vertices and the chromatic number of G, respectively. The first upper index (without brackets) will denote the number of vertices: G", S", T ",P are graphs of order n. Kph)(m r,..., m p) denotes the huniform hypergraph with m,+...+mp vertices partitioned into classes Cl,..., C p, where JQ=mi (i=1,..., p) and the hyperedges of this graph are those htuples, which have at most one vertex in each C i. For h=2 KP (ni l,..., nt p) is the ordinary complete ppartite graph. In some of our assertions we shall say e.g. that "changing o(nl) edges in G "... ". (Of course, o() cannot be applied to one graph.) As a matter of fact, in such cases we always consider a sequence of graphs G " and n.
DavenportSchinzel Theory Of Matrices
"... Let C be a configuration of 1's. We define f(n; C) to be the maximal number of 1's in a 01 matrix of size n \Theta n not having C as a subconfiguration. We consider the problem of determining the order of f(n; C) for several forbidden C's. Among others we prove that f(n; i 1 1 1 1 j ) = \Thet ..."
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Cited by 27 (1 self)
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Let C be a configuration of 1's. We define f(n; C) to be the maximal number of 1's in a 01 matrix of size n \Theta n not having C as a subconfiguration. We consider the problem of determining the order of f(n; C) for several forbidden C's. Among others we prove that f(n; i 1 1 1 1 j ) = \Theta(ff(n)n), where ff(n) is the inverse of the Ackermann function.
Extremal Problems on Set Systems
, 2002
"... For a family F (k) = fF 2 ; : : : ; F t g of kuniform hypergraphs let ex(n; F (k)) denote the maximum number of ktuples which a kuniform hypergraph on n vertices may have, while not containing any member of F (k). Let rk (n) denote the maximum cardinality of a set of integers Z [n], wh ..."
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Cited by 23 (14 self)
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For a family F (k) = fF 2 ; : : : ; F t g of kuniform hypergraphs let ex(n; F (k)) denote the maximum number of ktuples which a kuniform hypergraph on n vertices may have, while not containing any member of F (k). Let rk (n) denote the maximum cardinality of a set of integers Z [n], where Z contains no arithmetic progression of length k.
Blowup Lemma
 COMBINATORICA
, 1997
"... Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs. ..."
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Cited by 21 (2 self)
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Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs.
Turán's Extremal Problem In Random Graphs: Forbidding Odd Cycles
 J. Combin. Theory Ser. B
, 1995
"... For 0 0, there exists a real constant C = C(`; ), such that almost every random graph Gn;p with p = p(n) Cn 1+1=2` satisfies Gn;p ! 1=2+ C 2`+1 . In particular, for any fixed ` 1 and > 0, this result implies the existence of very sparse graphs G with G ! 1=2+ C 2`+1 . ..."
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Cited by 20 (7 self)
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For 0 0, there exists a real constant C = C(`; ), such that almost every random graph Gn;p with p = p(n) Cn 1+1=2` satisfies Gn;p ! 1=2+ C 2`+1 . In particular, for any fixed ` 1 and > 0, this result implies the existence of very sparse graphs G with G ! 1=2+ C 2`+1 .