Every set can been thought of as a directed graph whose edge relation is ∈. We show that many natural examples of directed graphs of this kind are indivisible: for every infinite κ, for every indecomposable λ, and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ℵ1 that are distinct (...) up to siblinghood. (shrink)

We investigate possible cardinalities of maximal antichains in the poset of copies \,\subseteq \rangle \) of a countable ultrahomogeneous relational structure \. It turns out that if the age of \ has the strong amalgamation property, then, defining a copy of \ to be large iff it has infinite intersection with each orbit of \, the structure \ can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (...) antichains of size continuum in the poset \\). Finally, we show that the posets of copies of all countable ultrahomogeneous partial orders contain maximal antichains of cardinality continuum and determine which of them contain countable maximal antichains. That holds, in particular, for the generic poset. (shrink)