We prove that if the universal minimal flow of a Polish group $G$ is metrizable and contains a $G_\delta$ orbit $G \cdot x_0$, then it is isomorphic to the completion of the homogeneous space $G/G_{ x_0}$ and show how this result translates naturally in terms of structural Ramsey theory. We also investigate universal minimal proximal flows and describe concrete representations of them in a number of examples.
Categories: math.DS math.FA math.GR math.LO