A topological group is extremely amenable if every continuous action of it on a compact Hausdorff space has a fixed point. We discuss a construction due to Uspenskij which gives a condition equivalent to extreme amenability for the setting of homeomorphism groups of compact metrizable spaces. We then show a Ramsey-type statement for subsets of simplices that, together with Uspenskij's construction, gives a new proof of a theorem due to Pestov: that the group $\textrm{Homeo}_+[0,1]$ is extremely amenable.

This is joint work with Lukas Michel and Alex Scott.