 
Summary: SIAM J. COMPUT. c 2006 Society for Industrial and Applied Mathematics
Vol. 36, No. 3, pp. 740762
RANDOM kSAT: TWO MOMENTS SUFFICE TO CROSS A SHARP
THRESHOLD
DIMITRIS ACHLIOPTAS AND CRISTOPHER MOORE
Abstract. Many NPcomplete constraint satisfaction problems appear to undergo a "phase tran
sition" from solubility to insolubility when the constraint density passes through a critical threshold.
In all such cases it is easy to derive upper bounds on the location of the threshold by showing that
above a certain density the first moment (expectation) of the number of solutions tends to zero. We
show that in the case of certain symmetric constraints, considering the second moment of the num
ber of solutions yields nearly matching lower bounds for the location of the threshold. Specifically,
we prove that the threshold for both random hypergraph 2colorability (Property B) and random
NotAllEqual kSAT is 2k1 ln 2  O(1). As a corollary, we establish that the threshold for random
kSAT is of order (2k), resolving a longstanding open problem.
Key words. satisfiability, random formulas, phase transitions
AMS subject classifications. Primary, 68R99, 82B26; Secondary, 05C80
DOI. 10.1137/S0097539703434231
1. Introduction. In the early 1900s, Bernstein [15] asked the following ques
tion: Given a collection of subsets of a set V , is there a partition of V into V1, V2 such
that no subset is contained in either V1 or V2? If we think of the elements of V as
