The Chvátal–Reed satisfiability threshold conjecture for random k-SAT

Let $F_k(n,m)$ denote the random $k$-SAT formula on $n$ variables with $m$ clauses, and let $\alpha=m/n$ be the clause density. Write $\alpha_\mathrm{sat}(k)$ for a putative critical density.

Chvátal–Reed conjecture. For all $k\geq 2$ there exists a value $\alpha_\mathrm{sat}(k)>0$ such that

$$\lim_{n\to\infty}\mathbb P(F_k(n,\alpha n)\text{ is satisfiable})=\begin{cases}1,&\text{if }\alpha<\alpha_\mathrm{sat}(k),\\\\0,&\text{if }\alpha>\alpha_\mathrm{sat}(k). \end{cases}$$

This asserts a sharp satisfiability phase transition, separating formulas that are satisfiable with high probability from those that are unsatisfiable with high probability. The case $k=2$ was proved in the cited work, while the general problem was presented as an ongoing research effort in the source.