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The Lovász Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Simplifications of his procedure and relaxations of its(More)
We construct binary codes for fingerprinting. Our codes for <i>n</i> users that are <i>&#949;</i>-secure against <i>c</i> pirates have length <i>O(c<sup>2</sup> log(n/&#949;))</i>. This improves the codes proposed by Boneh and Shaw [3] whose length is approximately the square of this length. Our codes are probabilistic. By proving matching lower bounds we(More)
This paper examines the extremal problem of how many 1-entries an n × n 0–1 matrix can have that avoids a certain fixed submatrix P. For any permutation matrix P we prove a linear bound, settling a conjecture of Zoltán Füredi and Péter Hajnal [8]. Due to the work of Martin Klazar [12], this also settles the conjecture of Stanley and Wilf on the number of(More)
Twenty years ago, Ajtai, Chvátal, Newborn, Szemerédi, and, independently, Leighton discovered that the crossing number of any graph with v vertices and e > 4v edges is at least ce 3 /v 2 , where c > 0 is an absolute constant. This result, known as the 'Crossing Lemma,' has found many important applications in discrete and computational geometry. It is tight(More)