On minimal actions of countable groups

Our purpose here is to review some recent developments in the theory of dynamical systems whose common theme is a link between minimal dynamical systems, certain Ramsey type combinatorial properties, and the Lovász local lemma (LLL). For a general countable group G the two classes of minimal systems we will deal with are (I) the minimal subsystems of the subgroup system (Sub(G),G), called URS’s (uniformly recurrent subgroups), and (II) minimal subshifts ; i.e. subsystems of the binary Bernoulli G-shift ({0,1}^G,{σg}_g∈G).