The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality $$\mathfrak {c}$$ c . We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality $$\mathfrak {c}$$ c , under different set-theoretic assumptions. We give a new proof of the old theorem of Harzheim, that the class of countably saturated linear orders has a uniquely determined one-element basis. From our proof it follows that this minimal linear order is a Fraïssé limit of certain Fraïssé class. In particular, it is homogeneous with respect to countable subsets. Next we prove the existence and uniqueness of the uncountable version of the random graph. This graph is isomorphic to $$(H(\omega _1),\in \cup \ni )$$ ( H ( ω 1 ) , ∈ ∪ ∋ ) , where $$H(\omega _1)$$ H ( ω 1 ) is the set of hereditarily countable sets, and two sets are connected if one of them is an element of the other. In the last section, an example of a prime countably saturated Boolean algebra is presented.

We study forcing notions similar to the Cohen forcing, which add some structures in given first-order language. These structures can be seen as versions of uncountable Fraïssé limits with finite conditions. Among them, we are primarily interested in linear orders.

We apply the technology developed in the 80s by Avraham, Rubin, and Shelah, to prove that the following is consistent with ZFC: there exists an uncountable, separable metric space X with rational distances, such that every uncountable partial 1-1 function from X to X is an isometry on an uncountable subset. This space contains a dense copy of the rational Urysohn space, and is homogeneous with respect to finite subspaces. We prove similar results for some other classes of structures, and in some cases show that the models we obtain are, in some sense, canonical.