In this paper, we introduce the enoughg-pairs property for principal coefficients cluster algebras, which can be understood as a strong version of the sign-coherence of theG-matrices. Then we prove that any skew-symmetrizable principal coefficients cluster algebra has the enoughg-pairs property. As applications, we prove some long standing conjectures in cluster algebras, including a conjecture on denominator vectors and a conjecture on exchange graphs (see Conjectures 1, 2 below). In addition, we give a criterion to distinguish whether particular cluster variables belong to one common cluster for any skew-symmetrizable cluster algebra. As a corollary, we prove by we prove a conclusion which was conjectured by Fomin et al., cf. (Acta Math 201(1):83-146, 2008, Conjecture 5.5) ."
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机译：在本文中，我们为主系数群集代数介绍了足够的足够的属性，可以理解为G矩阵的符号相干的强版本。 然后，我们证明了任何歪曲对称的主要系数簇代数具有足够的对象。 作为应用，我们在集群代数中证明了一些长时间的猜想，包括在额定参数矢量上的猜想和交换图上的猜想（参见下面的猜想1,2）。 此外，我们还提供了一个标准，以区分特定群集变量是否属于任何歪曲对称群集代数的一个公共集群。 作为一项必要性，我们证明了我们证明了由Fomin等人猜想的结论。，CF。 （Acta Math 201（1）：83-146,2008，猜想5.5）。“
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