The threshold for random k-SAT is 2k (ln 2 - O(k))

Let <i>F<sub>k</sub>(n,m)</i> be a random <i>k</i>-SAT formula on <i>n</i> variables formed by selecting uniformly and independently <i>m</i> out of all possible <i>k</i>-clauses. It is well-known that for <i>r ≥ 2<sup>k</sup> ln 2</i>, <i>F<sub>k</sub>(n,rn)</i> is unsatisfiable with probability <i>1-o(1)</i>. We prove that there exists a sequence <i>t<sub>k</sub> = O(k)</i> such that for <i>r ≥ 2<sup>k</sup> ln 2 - t<sub>k</sub></i>, <i>F<sub>k</sub>(n,rn)</i> is satisfiable with probability <i>1-o(1)</i>.Our technique yields an explicit lower bound for every <i>k</i> which for <i>k > 3</i> improves upon all previously known bounds. For example, when <i>k=10</i> our lower bound is 704.94 while the upper bound is 708.94.

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