Suppose we pick a 3-CNF with $n$ variables and $k$ clauses uniformly at random. How much is known about the probability $p(n,k)$ of being satisfiable?
2026-03-27 03:00:05.1774580405
Probability of a random CNF being satisfiable
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It is known that there is a threshold $r_c$ such that, for sequences $n_j, k_j \to \infty$ with $k_j/n_j \to r$, if $r > r_c$ then $p(n,k) \to 0$ while if $r < r_c$ then $p(n,k) \to 1$. Experimentally $r_c \approx 4.27$, but AFAIK a closed form for $r_c$ is not known. There are some rigorous upper and lower bounds; I don't know what the current best bounds are.