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

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.

• Did you try a reference search? cf. meta.cstheory.stackexchange.com/questions/300/… – RJK Sep 30 '10 at 15:15
• PS It looks to me like the second hit on Google is a freely-accessible article with answers to your question. (A 2009 arXiv article by Coja-Oghlan.) – RJK Sep 30 '10 at 15:23
• See cstheory.stackexchange.com/questions/1130/… for a reasonably up to date perspective. Follow-ups by the people working in this area, such as Amin Coja-Oghlan and Dimitris Achlioptas, then have the answer you are looking for. – András Salamon Sep 30 '10 at 15:39
• I still have not gotten a definite answer. I'll appreciate it if someone can give me a definite answer. Thanks – Tayfun Pay Oct 5 '10 at 22:13
• You may find this question useful: cstheory.stackexchange.com/questions/2168/…. In particular, see the first answer. – Giorgio Camerani Oct 25 '10 at 12:45

Dimitris Achlioptas covers this in his survey article from the Handbook of Satisfiability (PDF).

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.