Topological dynamics of unordered Ramsey structures

Moritz Müller¹ ; András Pongrácz²

¹ Kurt Gödel Research Center (KGRC) 1090 Wien, Austria
² Laboratoire d'Informatique (LIX) École Polytechnique 91128 Palaiseau, France

Fundamenta Mathematicae, Tome 230 (2015) no. 1, pp. 77-98.

Résumé

We investigate the connections between Ramsey properties of Fraïssé classes $\mathcal {K}$ and the universal minimal flow $M(G_\mathcal {K})$ of the automorphism group $G_\mathcal {K}$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class $\mathcal {K}$ has finite Ramsey degree for embeddings, then this degree equals the size of $M(G_\mathcal {K})$. We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if $\mathcal {K}$ is a relational Ramsey class and $G_\mathcal {K}$ is amenable, then $M(G_\mathcal {K})$ admits a unique invariant Borel probability measure that is concentrated on a unique generic orbit.

Détail

DOI : 10.4064/fm230-1-3

Keywords: investigate connections between ramsey properties fra classes mathcal universal minimal flow mathcal automorphism group mathcal their fra limits extension result kechris pestov todorcevic class mathcal has finite ramsey degree embeddings degree equals size mathcal partial answer question angel kechris lyons showing mathcal relational ramsey class mathcal amenable mathcal admits unique invariant borel probability measure concentrated unique generic orbit