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This paper is devoted to the study of universality for a particularcontinuous action naturally attached to certain pairs of closed subgroups of$S_{\infty}$. It shows that three new concepts, respectively called relativeextreme amenability, relative Ramsey property for embeddings, and relativeRamsey property for structures, are relevant in order to understand thisproperty correctly. It also allows to provide a partial answer to a questionposed by Kechris, Pestov and Todorcevic.

Universal flows of closed subgroups of $S_{\infty}$ and relative extreme amenability