A choice-free proof of the Erdős--Dushnik--Miller theorem

The Erdős–Dushnik–Miller theorem states that for every aleph $\kappa$, $$\kappa \to (\kappa ,\omega );$$ that is, every coloring $c:[\kappa {]}^2\to 2$ has either a $0$-homogeneous set of cardinality $\kappa$ or a $1$-homogeneous set of cardinality $\omega$. In this article, we present a purely combinatorial proof of this theorem in $\mathsf{Z}\mathsf{F}$ (i.e., Zermelo–Fraenkel set theory without the axiom of choice), avoiding any metamathematical considerations.