Random k-SAT: Two Moments Suffice to Cross a Sharp Threshold

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Many NP-complete constraint satisfaction problems appear to undergo a "phase transition'' from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal k-SAT is 2ᵏ⁻¹ ln 2 -O(1). As a corollary, we establish that the threshold for random k-SAT is of order Theta(2ᵏ), resolving a long-standing open problem.