The threshold for random k-SAT is 2 k (ln 2 - O(k))
Dimitris Achlioptas, Yuval Peres · 2003
Let Fk(n,m) be a random k-SAT formula on n variables formed by selecting uniformly and independently m out of all possible k-clauses. It is well-known that for r ≥ 2k ln 2, Fk(n,rn) is unsatisfiable with probability 1-o(1). We prove that there exists a sequence tk = O(k) such that for r ≥ 2k ln 2 - tk, Fk(n,rn) is satisfiable with probability 1-o(1).Our technique yields an explicit lower bound for every k which for k > 3 improves upon all previously known bounds. For example, when k=10 our lower bound is 704.94 while the upper bound is 708.94.