  In this paper, we introduce the enough $g$-pairs property for a principal
coefficients cluster algebra, which can be understood as a strong version of
the sign-coherence of the $G$-matrices. Then we prove that any
skew-symmetrizable principal coefficients cluster algebra has the enough
$g$-pairs property. As an application, we prove the positivity of denominator
vectors for any skew-symmetrizable cluster algebra. In fact, we give complete
answers to some long standing conjectures on denominator vectors of cluster
variables (see Conjecture 1.1 below), which are proposed by Fomin and
Zelevinsky in [Compos. Math. 143(2007), 112-164].
  In addition, we prove that the seeds whose clusters contain particular
cluster variables form a connected subgraph of the exchange graph of this
cluster algebra. Lastly, a criterion to distinguish whether particular cluster
variables belong to one common cluster is given.

Representation Theory;Commutative Algebra;Rings and Algebras