Finite presentation of homogeneous graphs, posets and Ramsey classes

It is commonly believed that one can prove Ramsey properties only for simple and '' well behaved '' structures. This is supported by the link of Ramsey classes of structures with homogeneous structures. We outline this correspondence in the context of the Classification Programme for Ramsey classes. As particular instances of this approach one can characterize all Ramsey classes of graphs, tournaments and partial ordered sets and also fully characterize all monotone Ramsey classes of relational systems (of any type). On the other side of this spectrum many homogeneous structures allow a concise description (called here a finite presentation) by means of all finite models of a suitable theory. Extending classical work of Rado (for the random graph) we find a finite presentation for each of the above classes where the classification problem is solved: (undirected) graphs, tournaments and partially ordered sets. The main result of the paper is a construction of classes P-is an element of and P-f of finite structures which are isomorphic to the generic (i.e. homogeneous and universal) partially ordered set. Somehow surprisingly, the structure P-is an element of extends Conway's surreal numbers and their linear ordering.