The threshold for random 𝑘-SAT is 2^{𝑘}log2-𝑂(𝑘)

Abstract

Let F k ( n , m ) F_k(n,m) be a random k k -CNF formula formed by selecting uniformly and independently m m out of all possible k k -clauses on n n variables. It is well known that if r ≥ 2 k log ⁡ 2 r \geq 2^k \log 2 , then F k ( n , r n ) F_k(n,rn) is unsatisfiable with probability that tends to 1 as n → ∞ n \to \infty . We prove that if r ≤ 2 k log ⁡ 2 − t k r \leq 2^k \log 2 - t_k , where t k = O ( k ) t_k = O(k) , then F k ( n , r n ) F_k(n,rn) is satisfiable with probability that tends to 1 as n → ∞ n \to \infty . Our technique, in fact, yields an explicit lower bound for the random k k -SAT threshold for every k k . For k ≥ 4 k \geq 4 our bounds improve all previously known such bounds.