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The purpose of this paper is to study the notion of relative extremeamenability for pairs of topological groups. We give a characterization by afixed point property on universal spaces. In addition we introduce the conceptsof an extremely amenable interpolant as well as maximally relatively extremelyamenable pairs and give examples. It is shown that relative extreme amenabilitydoes not imply the existence of an extremely amenable interpolant. The theoryis applied to generalize results of Kechris, Pestov and Todorcevic relating tothe application of Fra\"{i}ss\'e theory to the theory of Dynamical Systems. Inparticular, new conditions enabling to characterize universal minimal spaces ofautomorphism groups of Fra\"{i}ss\'e structures are given.

On relative extreme amenability