Dynamical properties of the Automorphism Groups of the Random Poset and Random Distributive Lattice Alexander S. Kechris and Miodrag Soki´ c 0 Introduction Let L be a countable first-order language. A class K of finite L-structures is called a Fra¨ ıss´ e class if it contains structures of arbitrarily large (finite) cardinality, is countable (in the sense that it contains only countably many isomorphism types) and satisfies the following: i) Hereditary property (HP): If B ∈K and A can be embedded in B, then A ∈K. ii) Joint Embedding Property (JEP): If A, B ∈K, there is C ∈K such that A, B can be embedded in C . iii) Amalgamation property (AP): If A, B, C ∈K and f : A → B,g : A → C are embeddings, there is D ∈K and embeddings r : B → D,s : C → D such that r ◦ f = s ◦ g. (Throughout this paper embeddings and substructures will be understood in the usual model theoretic sense (see, e.g., Hodges [Ho]); e.g., for graphs embeddings are induced embeddings, i.e., isomorphisms onto induced sub- graphs.) If K is a Fra¨ ıss´ e class, there is a unique, up to isomorphism, countably infinite structure K which is locally finite (i.e., finite generated substructures are finite), ultrahomogeneous (i.e., isomorphisms between finite substructures can be extended to automorphisms of the structure) and is such that, up to isomorphism, its finite substructures are exactly those in K. We call this the Fra¨ ıss´ e limit of K, in symbols K = Flim(K). 1