What is the number of $3$-CNF (conjunctive normal form) formulas with $n$ sentential variables and what is the fraction of satisfiable ones? I consider two formulas the same if they are syntactically the same modulo repeated clauses. Thus $$(x\_1\lor x\_1\lor \lnot x\_1)\text{ and } (x\_1\lor x\_1\lor \lnot x\_1)\land (x\_1\lor x\_1\lor \lnot x\_1)$$ are the same while $$(x\_1\lor x\_1\lor \lnot x\_1)\text{ and } (x\_1\lor x\_1\lor \lnot x\_1)\land (x\_1\lor \lnot x\_1\lor x\_1)$$ are **not**. Now, what is the number of these equivalence classes of formulas in $n$ variables and what is the fraction of satisfiable ones? If the exact numbers cannot be given, just some recurrence formulas will do. Does the $\lim$ of the fraction of satisfiable ones go to $0$ as $n\to \infty$?



The number of $3$-CNF formulas in $n$-variables and the fraction of satisfiable ones
Regarding the fraction of satisfiable 3-CNF formulas in $n$ variables, it is widely believed that there is a phase transition that occurs depending on how many clauses there are compared to the number of variables. To be precise, it is conjectured (but not yet proven) that if there are more than $\alpha n$ clauses, then the formula is almost surely unsatisfiable, and if there are less than $\alpha n$ clauses then the formula is almost surely satisfiable (where $\alpha$ is around 4.2667). This phase transition has been established for $k$-SAT when $k$ is large. See [Proof of the satisfiability conjecture for large $k$](https://arxiv.org/abs/1411.0650) by Ding, Sly, and Sun.


