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The Lovász Sieve

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Extremal Combinatorics

Part of the book series: Texts in Theoretical Computer Science. An EATCS Series ((TTCS))

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Abstract

Assume that we have a family of “bad” events. How can we make sure that there is some non-zero probability that none of the bad events will happen? By the union bound Pr[AB]≤Pr[A]+Pr[B], this probability is non-zero if the sum of probabilities of all these bad events is smaller than 1. In one sense this is best possible: when bad events are pairwise disjoint, the condition cannot be weakened. If we know that the bad events are independent, we can get a much better bound, by multiplying all the probabilities that each single bad event does not happen. This will work as long as each bad event has probability smaller than 1. But this will immediately fail, if at least two of the bad events are not independent.

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Correspondence to Stasys Jukna .

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© 2011 Springer-Verlag Berlin Heidelberg

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Jukna, S. (2011). The Lovász Sieve. In: Extremal Combinatorics. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17364-6_19

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  • DOI: https://doi.org/10.1007/978-3-642-17364-6_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-17363-9

  • Online ISBN: 978-3-642-17364-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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