The Threshold for Random k-SAT is 2ᵏ ln2 - O(k)

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