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An ergodic Szemer'edi theorem for commuting transformations
 J. Analyse Math
, 1979
"... The classical Poincar6 recurrence theorem asserts that under the action of a measure preserving transformation T of a finite measure space (X, ~, p.), every set A of positive measure recurs in the sense that for some n> 0,/z (T'A n A)> 0. In [1] this was extended to multiple recurrence: ..."
Abstract

Cited by 113 (2 self)
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: the transformations T, T2,..., T k have a common power satisfying /x (A n ThA n... n Tk"A)> 0 for a set A of positive measure. We also showed that this result implies Szemer6di's theorem stating that any set of integers of positive upper density contains arbitrarily long arithmetic progressions. In [2
On graphs with linear Ramsey numbers
 J. GRAPH THEORY
, 2000
"... For a fixed graph H, the Ramsey number r (H) is defined to be the least integer N such that in any 2coloring of the edges of the complete graph KN, some monochromatic copy of H is always formed. Let H(n,) denote the class of graphs H having n vertices and maximum degree at most. It was shown by Chv ..."
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Cited by 34 (2 self)
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by ChvataÂl, Rödl, SzemereÂ di, and Trotter that for each there exists c ( ) such that r (H) < c ()n for all H 2H(n,). That is, the Ramsey numbers grow linearly with the size of H. However, their proof relied on the wellknown regularity lemma of SzemereÂ di and only gave an upper bound for c ( ) which
unknown title
"... let P denote the processor bound, and T denote the time bound ofa parallel algorithm for a given problem, the product PT is, clearly, lower bounded by the minimum sequential time, Ts, required to solve this problem. We say a parallel algorithm is optimal ifPT O(Ts). Discovering optimal parallel algo ..."
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is indeed optimal. Simultaneously, Atai, Koml6s, and Szemer6di [4] discovered a deterministic parallel algorithm for sorting n general keys in time O(log n) using a sorting network of O(n log n) processors. Later, Leighton [17] showed that this algorithm could be modified to run in O(log n) time on an n