In this artcile, we firstly prove that any skew-symmetrizable cluster algebra with principal coefficients at $t_0$ has the enough $g$-pairs property. The proof depends on a result by Muller on scattering diagrams. Based on this property of cluster algebras, we give a new explanation of the sign-coherence of $G$-matrices, and a positive answer to the conjecture on denominator vectors proposed by Fomin and Zelevinsky. Moreover, we give two applications of the positive answer. As the first application, any skew-symmetrizable cluster algebra $\mathcal A(\mathcal S)$ is proved to have the proper Laurent monomial property. In the viewpoint of a theorem by Irelli and Labardini-Fragoso, we actually provide an unified method to prove the linear independence of cluster monomials of $\mathcal A(\mathcal S)$. Secondly, by the positive answer said above, a function which is called compatibility degree can be well-defined on the set of cluster variables of $\mathcal A(\mathcal S)$, and then a characterization for several cluster variables to be contained or not in some cluster of $\mathcal A(\mathcal S)$ is given via compatibility degree, which is the other application of the positive answer.

中文翻译：

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集群代数的足够g-pairs属性和分母向量

在本文中，我们首先证明任何主系数为 $t_0$ 的偏对称簇代数具有足够的 $g$-pairs 属性。证明取决于穆勒在散射图上的结果。基于簇代数的这一性质，我们对$G$矩阵的符号相干性给出了新的解释，并对Fomin和Zelevinsky提出的分母向量猜想给出了肯定的答案。此外，我们给出了肯定答案的两种应用。作为第一个应用，任何偏斜对称的簇代数 $\mathcal A(\mathcal S)$ 被证明具有适当的 Laurent 单项式性质。从 Irelli 和 Labardini-Fragoso 的一个定理的角度来看，我们实际上提供了一个统一的方法来证明 $\mathcal A(\mathcal S)$ 的簇单项式的线性独立性。其次，通过上面所说的肯定回答，