What are the current best known upper and lower bounds on the (un)satisfiability threshold for random k-sat and/or 3-sat?

Question

I would like to know the current state of the phase transition for random k-sat, given n variables and m clauses, what is the best known c=m/n for upper and lower bounds.

Answer 1

Dimitris Achlioptas covers this in a survey article from the first edition of the Handbook of Satisfiability (PDF of draft).

There is conjectured to be a single threshold $r_k$ for each $k \ge 3$, so that when $m/n > r_k$ then a random $k$-SAT formula with $m$ clauses and $n$ variables is unsatisfiable with high probability, and so that when $m/n < r_k$ then a random $m$-clause, $n$-variable $k$-SAT formula is satisfiable with high probability. (More precisely, the conjecture is that in the limit as $n$ tends to infinity, the probability of satisfiability is 0 or 1 in these two regimes, respectively.)

Assuming that this Satisfiability Threshold Conjecture holds, the best known bounds for $r_k$ are

k                   3      4      5     7     10      20
Best upper bound 4.51  10.23  21.33 87.88 708.94 726,817
Best lower bound 3.52   7.91  18.79 84.82 704.94 726,809

(this table appears on the page indicated as 247 in the draft).

In a more recent manuscript (arXiv:1411.0650), Jian Ding, Allan Sly and Nike Sun showed that for all sufficiently large $k$, there is in fact a single threshold $r_k = 2^k\ln 2 - (1+\ln 2)/2 + o(1)$, and the error term $o(1)$ in this expression goes to zero as $k$ tends to infinity.