Indivisible sets and well‐founded orientations of the Rado graph

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: H κ for every infinite κ, V λ 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 2 ℵ 0indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ℵ 1 that are distinct up to siblinghood.