Ramsey classes with forbidden homomorphisms and a closure

Extending classical early Ramsey-type results, the structural Ramsey theory originated at the beginning of 70ies, see [11] for references. However the list of Ramsey classes, as the top of the line of Ramsey properties, was somewhat limited. This was also supported by result of Nešetřil [7] that made a connection between Ramsey classes and ultrahomogeneous structures showing particularly that there are only four types of Ramsey classes of undirected graphs. This connection led to the classification programme for Ramsey classes [8] and, perhaps more importantly, to the connection to the topological dynamics and ergodic theory [6].