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14
On the Evolution of Random Graphs
 PUBLICATION OF THE MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES
, 1960
"... his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_ ..."
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Cited by 1849 (7 self)
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his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_
PseudoRandom Graphs
 IN: MORE SETS, GRAPHS AND NUMBERS, BOLYAI SOCIETY MATHEMATICAL STUDIES 15
"... ..."
On Brooks' theorem for sparse graphs
 Combinatorics, Probability and Computing
, 1995
"... Let G be a graph with maximum degree ∆(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfies χ(G) ≤ (1 + o(1)) ∆(G) log ∆(G) where o(1) goes to zero as ∆(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same ..."
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Cited by 35 (4 self)
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Let G be a graph with maximum degree ∆(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfies χ(G) ≤ (1 + o(1)) ∆(G) log ∆(G) where o(1) goes to zero as ∆(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same bound for the listchromatic (or choice) number: provided g(G)> 4. 1
Flag algebras
 Journal of Symbolic Logic
"... Abstract. Asymptotic extremal combinatorics deals with questions that in the language of model theory can be restated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a random ..."
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Cited by 31 (3 self)
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Abstract. Asymptotic extremal combinatorics deals with questions that in the language of model theory can be restated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a randomly chosen submodel of N with M  elements is isomorphic to M. Which asymptotic relations exist between the quantities p(M1, N),..., p(Mh, N), where M1,..., Mh are fixed “template ” models and N  grows to infinity? In this paper we develop a formal calculus that captures many standard arguments in the area, both previously known and apparently new. We give the first application of this formalism by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density. §1. Introduction. A substantial part of modern extremal combinatorics (which will be called here asymptotic extremal combinatorics) studies densities with which some “template ” combinatorial structures may or may not appear in unknown (large) structures of the same type1. As a typical example, let Gn be a
Improved Upper Bounds on Stopping Redundancy
, 2007
"... For a linear block code with minimum distance d, its stopping redundancy is the minimum number of check nodes in a Tanner graph representation of the code, such that all nonempty stopping sets have size d or larger. We derive new upper bounds on stopping redundancy for all linear codes in general, ..."
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Cited by 21 (3 self)
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For a linear block code with minimum distance d, its stopping redundancy is the minimum number of check nodes in a Tanner graph representation of the code, such that all nonempty stopping sets have size d or larger. We derive new upper bounds on stopping redundancy for all linear codes in general, and for maximum distance separable (MDS) codes specifically, and show how they improve upon previous results. For MDS codes, the new bounds are found by upperbounding the stopping redundancy by a combinatorial quantity closely related to Turán numbers. (The Turán number, „
On Turán’s theorem for sparse graphs
 Combinatorica
, 1981
"... For a graph G with n vertices and average valency t, Turán's theorem yields the inequality a?nl(t+1) where a denotes the maximum size of an independent set in G. We improve this bound for graphs containing no large cliques. 0. Notation n=n(G)=number of vertices of the graph G e=e(G)=number of edges ..."
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Cited by 10 (0 self)
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For a graph G with n vertices and average valency t, Turán's theorem yields the inequality a?nl(t+1) where a denotes the maximum size of an independent set in G. We improve this bound for graphs containing no large cliques. 0. Notation n=n(G)=number of vertices of the graph G e=e(G)=number of edges of G h=h(G)=number of triangles in G deg (P)=valency (degree) of the vertex P deg, (P) = trianglevalency of P=number of triangles in G adjacent to P t=t(G)=n f deg (P) = 2eln = average valency in G (we will tacitly assume t ' 1) T = T(G) =maximum valency in G a=a(G)=maximum size of independent set of vertices (independence or stability number) Kp =shorthana ' for pclique log x=max {1, In x} to, cl, c2,... are absolute constants when speaking of union, difference or partition of graphs, we work with the vertexsets Let G be a graph of n vertices and a edges with average valency t=2eln. It is an easy consequence of the celebrated Turán's theorem [6] (and can easily be proved directly) that G contains an independent set of size nl(t+1), i.e.
On 3hypergraphs with forbidden 4vertex configurations
, 2010
"... Every 3graph in which no four vertices are independent and no four vertices span precisely three edges must have edge density ≥ 4/9(1 − o(1)). This bound is tight. The proof is a rather elaborate application of CauchySchwarz type arguments presented in the framework of flag algebras. We include fu ..."
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Cited by 9 (1 self)
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Every 3graph in which no four vertices are independent and no four vertices span precisely three edges must have edge density ≥ 4/9(1 − o(1)). This bound is tight. The proof is a rather elaborate application of CauchySchwarz type arguments presented in the framework of flag algebras. We include further demonstrations of this method by reproving a few known tight results about hypergraph Turán densities and significantly improving numerical bounds for several problems for which the exact value is not known yet.
A generalization of Turán's theorem
"... In this paper we obtain an asymptotic generalization of Turán's theorem. We prove that if all the nontrivial eigenvalues of a dregular graph G on n vertices are sufficiently small, then the largest K t free subgraph of G contains approximately t 1 fraction of its edges. Turán's theorem correspon ..."
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Cited by 5 (1 self)
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In this paper we obtain an asymptotic generalization of Turán's theorem. We prove that if all the nontrivial eigenvalues of a dregular graph G on n vertices are sufficiently small, then the largest K t free subgraph of G contains approximately t 1 fraction of its edges. Turán's theorem corresponds to the case d = n 1.
Embedding complete trees into the hypercube
 DISCRETE APPL. MATH
, 2001
"... We consider embeddings of the complete tary trees of depth k (denotation T k,t) as subgraphs into the hypercube of minimum dimension n. This n, denoted by dim(T k,t), is known if max{k, t} ≤ 2. First we study the next open case max{k, t} = 3. We improve the known upper bound dim(T k,3) ≤ 2k + 1 ..."
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Cited by 4 (0 self)
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We consider embeddings of the complete tary trees of depth k (denotation T k,t) as subgraphs into the hypercube of minimum dimension n. This n, denoted by dim(T k,t), is known if max{k, t} ≤ 2. First we study the next open case max{k, t} = 3. We improve the known upper bound dim(T k,3) ≤ 2k + 1 up to limk→ ∞ dim(T k,3)/k ≤ 5/3 and derive the asymptotic limt→ ∞ dim(T 3,t)/t = 227/120. As a coresult, we present an exact formula for the dimension of arbitrary trees of depth 2, as a function of their vertex degrees. These results and new techniques provide an improvement of the known upper bound for dim(T k,t) for arbitrary k and t.
Turán’s theorem in sparse random graphs
 MR
, 2003
"... We prove the analogue of Turán’s Theorem in random graphs with edge probability p(n) ≫ n−1/(k−1.5) 1. With probability 1 − o(1), one needs to delete approximately k−1 –fraction of the edges in a random graph in order to destroy all cliques of size k. 1 ..."
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Cited by 3 (0 self)
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We prove the analogue of Turán’s Theorem in random graphs with edge probability p(n) ≫ n−1/(k−1.5) 1. With probability 1 − o(1), one needs to delete approximately k−1 –fraction of the edges in a random graph in order to destroy all cliques of size k. 1