We represent the universal Menger curve as the topological realization |𝕄| of the projective Fraïssé limit 𝕄 of the class of all finite connected graphs. We show that 𝕄 satisfies combinatorial analogues of the Mayer–Oversteegen–Tymchatyn homogeneity theorem and the Anderson–Wilson projective universality theorem. Our arguments involve only 0-dimensional topology and constructions on finite graphs. Using the topological realization 𝕄↦|𝕄|, we transfer some of these properties to the Menger curve: we prove the approximate projective homogeneity theorem, recover Anderson’s finite homogeneity theorem, and prove a variant of Anderson–Wilson’s theorem. The finite homogeneity theorem is the first instance of an “injective” homogeneity theorem being proved using the projective Fraïssé method. We indicate how our approach to the Menger curve may extend to higher dimensions.

中文翻译：

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门格尔曲线的组合模型

我们将通用门格尔曲线表示为拓扑实现|𝕄|射影 Fraïssé 极限𝕄的所有有限连通图的类。我们表明𝕄满足 Mayer-Oversteegen-Tymchatyn 同质性定理和 Anderson-Wilson 射影普遍性定理的组合类​​似物。我们的论点仅涉及有限图上的 0 维拓扑和构造。使用拓扑实现𝕄↦|𝕄|，我们将其中一些性质转移到门格尔曲线：我们证明了近似射影同质性定理，恢复了安德森有限同质性定理，并证明了安德森-威尔逊定理的一个变体。有限同质性定理是使用射影 Fraïssé 方法证明的“内射”同质性定理的第一个实例。我们指出我们对门格尔曲线的方法如何扩展到更高的维度。