The Asymptotic Order of the Random k -SAT Threshold

Form a random k-SAT formula on n variables by selecting uniformly and independently m = rn clauses out of all 2 ( n k ) possible k-clauses. The Satisfiability Threshold Conjecture asserts that for each k there exists a constant rk such that, as n tends to infinity, the probability that the formula is satisfiable tends to 1 if r < rk and to 0 if r > rk . It has long been known that 2/k < rk < 2. We prove that rk > 2 k−1 ln 2 − dk, where dk → (1 + ln 2)/2. Our proof also allows a blurry glimpse of the “geometry” of the set of satisfying truth assignments.