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66
Books in graphs
, 2008
"... A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) n 2 wi ..."
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Cited by 1797 (20 self)
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A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) n 2 with 0 < α < 17 −3 (), and G has no book of size at least graph G1 of order at least
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 204 (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...
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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Cited by 23 (0 self)
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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.
THE TURÁN NUMBER OF THE FANO PLANE
 COMBINATORICA 25 (5) (2005) 561–574
, 2005
"... Let PG2(2) be the Fano plane, i.e., the unique hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. In this paper we prove that for sufficiently large n, the maximum number of edges in a 3uniform hypergraph on n vertices not containing a Fano plan ..."
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Cited by 17 (1 self)
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Let PG2(2) be the Fano plane, i.e., the unique hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. In this paper we prove that for sufficiently large n, the maximum number of edges in a 3uniform hypergraph on n vertices not containing a Fano plane is ex � n, P G2(2) � = n
The number of edge colorings with no monochromatic cliques
 J. London Math. Soc
, 2004
"... Let F (n, r, k) denote the maximum possible number of distinct edgecolorings of a simple graph on n vertices with r colors which contain no monochromatic copy of Kk. Itisshownthatfor every fixed k and all n>n0(k), F (n, 2,k)=2 t k −1(n) and F (n, 3,k)=3 t k −1(n),wheretk−1(n) is the maximum possibl ..."
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Cited by 13 (5 self)
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Let F (n, r, k) denote the maximum possible number of distinct edgecolorings of a simple graph on n vertices with r colors which contain no monochromatic copy of Kk. Itisshownthatfor every fixed k and all n>n0(k), F (n, 2,k)=2 t k −1(n) and F (n, 3,k)=3 t k −1(n),wheretk−1(n) is the maximum possible number of edges of a graph on n vertices with no Kk (determined by Turán’s theorem). The case r = 2 settles an old conjecture of Erdős and Rothschild, which was also independently raised later by Yuster. On the other hand, for every fixed r>3andk>2, the function F (n, r, k) is exponentially bigger than r t k −1(n). The proofs are based on Szemerédi’s regularity lemma together with some additional tools in extremal graph theory, and provide one of the rare examples of a precise result proved by applying this lemma. 1.
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 12 (9 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
Finitely forcible graphons
 arXiv:0901.0929. BÉLA BOLLOBÁS, SVANTE JANSON, AND
, 2009
"... We investigate families of graphs and graphons (graph limits) that are defined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities. Generalizing results of Tu ..."
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Cited by 10 (0 self)
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We investigate families of graphs and graphons (graph limits) that are defined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities. Generalizing results of Turán, Erdős–Simonovits and Chung–Graham–Wilson, we construct numerous finitely forcible graphons. Most of these fall into two categories: one type has an algebraic structure and the other type has an iterated (fractallike) structure. We also give some necessary conditions for forcibility, which imply that finitely forcible graphons are “rare”, and exhibit simple and explicit nonforcible graphons.