# 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, 2010 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, 2010 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. Sep 30, 2010 at 15:39
• You may find this question useful: cstheory.stackexchange.com/questions/2168/…. In particular, see the first answer. Oct 25, 2010 at 12:45
• @Tayfun Pay: I am commenting here because you have deleted your questions and I can't comment there. Sorry, I didn't meant to be rude, I was asking for motivation to understand why you are interested in the question. In the other case, if you write down the definitions you will see the difference between them (many-one reductions does not need to preserve the number of certificates, it can map an instance of the problem A to an instance of problem B and there is no condition on the certificates at all). I think it would be better if you migrated the question to Math.SE. (more) Aug 3, 2011 at 2:18

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.

• Unfortunately in the 2nd edition of the Handbook this table is unpopulated (it contains some placeholders but no actual values). Oct 20, 2021 at 10:39
• I suppose this is for when all clauses have width exactly $k$. Jul 8 at 20:08