Ramsey Properties of Finite Posets

An important problem in topological dynamics is the calculation of the universal minimal flow of a topological group. When the universal minimal flow is one point, we say that the group is extremely amenable. For the automorphism group of Fra?ssé structures, this problem has been translated into a question about the Ramsey and ordering properties of certain classes of finite structures by Kechris et al. (Geom Funct Anal 15:106–189, 2005). Using the Schmerl list (Schmerl, Algebra Univers 9:317–321, 1979) of Fra?ssé posets, we consider classes of finite posets with arbitrary linear orderings and linear orderings that are linear extensions of the partial ordering. We provide classification of each of these classes according to their Ramsey and ordering properties. Additionally, we extend the list of extremely amenable groups as well as the list of metrizable universal minimal flows.