Three forms of the Erdős–Dushnik–Miller theorem

We continue the study of the Erdős–Dushnik–Miller theorem (A graph with an uncountable set of vertices has either an infinite independent set or an uncountable clique) in set theory without the axiom of choice. We show that there are three inequivalent versions of this theorem and we give some results about the positions of these versions in the deductive hierarchy of weak choice principles. Furthermore, we settle some open problems from Tachtsis [Monatsh. Math. 203 (2024), 677–693] and from Banerjee and Gopaulsingh [Bull. Polish Acad. Sci. Math. 71 (2023), 1–21].