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Abstract

Let $\mathcal {K}$ be a class of finite relational structures. We define $\mathcal {EK}$ to be the class of finite relational structures $\mathbf {A}$ such that $\mathbf {A}/E\in \mathcal {K}$, where $E$ is an equivalence relation defined on the structure $\mathbf {A}$. Adding arbitrary linear orderings to structures from $\mathcal {EK}$, we get the class $\mathcal {OEK}$. If we add linear orderings to structures from $\mathcal {EK}$ such that each $E$-equivalence class is an interval then we get the class $\mathcal {CE}[\mathcal{K}^{\ast }]$. We provide a list of Fraïssé classes among $\mathcal {EK}$, $\mathcal {OEK}$ and $\mathcal {CE}[\mathcal{K}^{\ast }]$. In addition, we classify $\mathcal {OEK}$ and $\mathcal {CE}[\mathcal{K}^{\ast }]$ according to the Ramsey property. We also conduct the same analysis after adding additional structure to each equivalence class. As an application, we give a topological interpretation using the technique introduced in Kechris, Pestov and Todorčević. In particular, we extend the lists of known extremely amenable groups and universal minimal flows.