Ramsey Properties of Permutations

The age of each countable homogeneous permutation forms a Ramsey class. Thus, there are five countably infinite Ramsey classes of permutations.Comment: 10 pages, 3 figures; v2: updated info on related work + some other minor enhancements (Dec 21, 2012

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

This paper is devoted to the study of universality for a particular continuous action naturally attached to certain pairs of closed subgroups of $S_{\infty}$. It shows that three new concepts, respectively called relative extreme amenability, relative Ramsey property for embeddings, and relative Ramsey property for structures, are relevant in order to understand this property correctly. It also allows to provide a partial answer to a question posed by Kechris, Pestov and Todorcevic.Comment: 13 pages. Updated references, some statements made more precise. arXiv admin note: text overlap with arXiv:1201.127

Polish groups with metrizable universal minimal flows

International audienceWe prove that if the universalminimal flow of a Polish group G is metrizable and contains a comeagre orbit O, then it is isomorphic to the completion of the homogeneous space G/O 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