Ramsey classes and partial orders

We consider the classes of finite coloured partial orders, i.e., partial orders together with unary relations determining the colour of their points. These classes are the ages of the countable homogeneous coloured partial orders, classified by Torrezão de Souza and Truss in 2008. We prove that certain classes can be expanded with an order to become Ramsey classes with the ordering property. The motivation for finding such classes is the 2005 paper of Kechris, Pestov and Todorcevic, showing that these concepts are important in topological dynamics for calculating universal minimal flow of automorphism groups of homogeneous structures and finding new examples of extremely amenable groups.

We introduce the elementary skeletons to enumerate the classes of ordered shaped partial orders and show that classes are Ramsey using three main approaches. With the Blowup Lemma we use the known results about the Ramsey classes of ordered partial orders, to prove results about shaped classes. We use the Structural Product Ramsey Lemma to show that a class K is Ramsey when structures in classes known to be Ramsey determine each structure in K uniquely. Finally, we use the Two Pass Lemma when each structure in the considered class has two dimensions that can be built separately and the classes corresponding to both dimensions of the structure are known to be Ramsey. We then show that the classes of unordered reducts of the structures in the classes enumerated by elementary skeletons are the Fraïssé limits of the countable homogeneous coloured partial orders.