In a recent paper by a superset of the authors it was proved that for every primitive 3-constrained space $\Gamma$ of finite diameter $\delta$ from Cherlin's catalogue of metrically homogeneous graphs, there exists a finite family $\mathcal F$ of $\{1,\ldots, \delta\}$-edge-labelled cycles such that a $\{1,\ldots, \delta\}$-edge-labelled graph is a subgraph of $\Gamma$ if and only if it contains no homomorphic images of cycles from $\mathcal F$. However, the cycles in the families $\mathcal F$ were not described explicitly as it was not necessary for the analysis of Ramsey expansions and the extension property for partial automorphisms. This paper fills this gap by providing an explicit description of the cycles in the families $\mathcal F$, heavily using the previous result in the process. Additionally, we explore the potential applications of this result, such as interpreting the graphs as semigroup-valued metric spaces or homogenizations of $\omega$-categorical $\{1,\delta\}$-edge-labelled graphs.

翻译：在最近一篇由本文作者超集所撰写的论文中，证明了对于Cherlin度量齐次图目录中每个具有有限直径$\delta$的原始3-约束空间$\Gamma$，存在一个有限的$\{1,\ldots, \delta\}$-边标记环族$\mathcal F$，使得一个$\{1,\ldots, \delta\}$-边标记图是$\Gamma$的子图当且仅当其不包含$\mathcal F$中任何环的同态像。然而，由于当时对于Ramsey扩张和部分自同构扩展性质的分析无需显式描述，这些环族$\mathcal F$中的环并未被具体刻画。本文通过大量利用先前结果，填补了这一空白，给出了$\mathcal F$中环的显式描述。此外，我们还探讨了该结果的潜在应用，例如将图解释为半群值度量空间或$\omega$-范畴化$\{1,\delta\}$-边标记图的齐次化。