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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_
Szemerédi’s regularity lemma and quasirandomness
 CMS BOOKS MATH./OUVRAGES MATH. SMC
, 2003
"... The first half of this paper is mainly expository, and aims at introducing the regularity lemma of Szemerédi. Among others, we discuss an early application of the regularity lemma that relates the notions of universality and uniform distribution of edges, a form of ‘pseudorandomness’ or ‘quasirand ..."
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Cited by 17 (9 self)
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The first half of this paper is mainly expository, and aims at introducing the regularity lemma of Szemerédi. Among others, we discuss an early application of the regularity lemma that relates the notions of universality and uniform distribution of edges, a form of ‘pseudorandomness’ or ‘quasirandomness’. We then state two closely related variants of the regularity lemma for sparse graphs and present a proof for one of them. In the second half of the paper, we discuss a basic idea underlying the algorithmic version of the original regularity lemma: we discuss a ‘local’ condition on graphs that turns out to be, roughly speaking, equivalent to the regularity condition of Szemerédi. Finally, we show how the sparse version of the regularity lemma may be used to prove the equivalence of a related, local condition for regularity. This new condition turns out to give a O(n²) time algorithm for testing the quasirandomness of an nvertex graph.
Regular pairs in sparse random graphs I
 RANDOM STRUCTURES ALGORITHMS
, 2003
"... We consider bipartite subgraphs of sparse random graphs that are regular in the sense of Szemeredi and, among other things, show that they must satisfy a certain local pseudorandom property. This property and its consequences turn out to be useful when considering embedding problems in subgraphs of ..."
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Cited by 17 (8 self)
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We consider bipartite subgraphs of sparse random graphs that are regular in the sense of Szemeredi and, among other things, show that they must satisfy a certain local pseudorandom property. This property and its consequences turn out to be useful when considering embedding problems in subgraphs of sparse random graphs.
unknown title
, 2009
"... How to obtain sharp threshold functions for the random intersection graph via coupling method? ..."
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How to obtain sharp threshold functions for the random intersection graph via coupling method?