Current tightest bounds for critical 3-SAT density 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 $3.52&lt;\alpha&lt;4.51$ if $\alpha$ exists. 

What are the best known bounds for $\alpha$?
Notwithstanding Friedgut's theorem about $k$-SAT, while we lack techniques to get to negligible $\epsilon$ for small $k$, it seems more useful to talk about the satisfiability threshold ($\alpha - \epsilon$) and the unsatisfiability threshold ($\alpha + \epsilon$) as separate entities.

The unsatisfiability threshold is known to be at most 4.4898, a slight improvement since Maneva's 2001 thesis.


J. Díaz, L. Kirousis, D. Mitsche, X. Pérez-Gimenéz.  On the satisfiability threshold of formulas with three literals per clause, Theoretical Computer Science 410, 2009, 2920–2934. doi:10.1016/j.tcs.2009.02.020 (preprint from one of the authors)


The satisfiability threshold is known to be at least 3.52, which is unchanged from the time of Maneva's thesis.


A. C. Kaporis, L. M. Kirousis, E. G. Lalas.  The Probabilistic Analysis of a Greedy Satisfiability Algorithm, Random Structures and Algorithms 28, 2006, 444–480.  doi:10.1002/rsa.20104


These bounds were recently cited by Achlioptas and Menchaca-Mendez as the best known to date.


D. Achlioptas, R. Menchaca-Mendez.  Unsatisfiability Bounds for Random CSPs from an Energetic Interpolation Method, ICALP 2012, LNCS 7391, 1–12.  doi:10.1007/978-3-642-31594-7_1