I'm interested in the critical 3-satisfiability (3-SAT) density $\alpha$. It's conjectured that such $\alpha$ exists: if the number of randomly generated 3-SAT clauses is $(\alpha + \epsilon) n$ or more, they are almost surely unsatisfiable. (Here $\epsilon$ is any small constant and $n$ is the number of variables.) If the number is $(\alpha - \epsilon) n$ or less, they are almost surely satisfiable.
The thesis Belief propagation algorithms for constraint satisfaction problems by Elitza Nikolaeva Maneva challenges the problem from the angle of belief propagation known in information theory. On page 13, it says $3.52<\alpha<4.51$ if $\alpha$ exists.
What are the best known bounds for $\alpha$?