Indivisible sets and well‐founded orientations of the Rado graph

Every set can been thought of asa directed graph whose edge relationis . We show that many natural examplesof directed graphs of this kindare indivisible: H for everyinfinite , V for everyindecomposable , and every countable modelof settheory. All of the countable digraphs weconsider are orientations of the countablerandom graph. In this way we find 20 indivisible well-founded orientationsof the random graph that are distinct upto isomorphism, and (1) thatare distinct up tosiblinghood.