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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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The threshold for random $k$-SAT is $2^k\log 2-O(k)$
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by Dimitris Achlioptas and Yuval Peres PDF
J. Amer. Math. Soc. 17 (2004), 947-973 Request permission


Let $F_k(n,m)$ be a random $k$-CNF formula formed by selecting uniformly and independently $m$ out of all possible $k$-clauses on $n$ variables. It is well known that if $r \geq 2^k \log 2$, then $F_k(n,rn)$ is unsatisfiable with probability that tends to 1 as $n \to \infty$. We prove that if $r \leq 2^k \log 2 - t_k$, where $t_k = O(k)$, then $F_k(n,rn)$ is satisfiable with probability that tends to 1 as $n \to \infty$. Our technique, in fact, yields an explicit lower bound for the random $k$-SAT threshold for every $k$. For $k \geq 4$ our bounds improve all previously known such bounds.
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Additional Information
  • Dimitris Achlioptas
  • Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052
  • Email:
  • Yuval Peres
  • Affiliation: Department of Statistics, University of California, Berkeley, California 94720
  • MR Author ID: 137920
  • Email:
  • Received by editor(s): September 4, 2003
  • Published electronically: August 27, 2004
  • Additional Notes: This research was supported by NSF Grant DMS-0104073, NSF Grant DMS-0244479 and a Miller Professorship at UC Berkeley. Part of this work was done while visiting Microsoft Research.
  • © Copyright 2004 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 17 (2004), 947-973
  • MSC (2000): Primary 68R99, 82B26; Secondary 05C80
  • DOI:
  • MathSciNet review: 2083472