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

Abstract

The Erdös--Dushnik--Miller theorem states that for every aleph $κ$, \[ κ\to(κ,ω); \] that is, every coloring $c:[κ]^2\to2$ has either a $0$-homogeneous set of cardinality $κ$ or a $1$-homogeneous set of cardinality $ω$. In this article, we present a purely combinatorial proof of this theorem in $\mathsf{ZF}$ (i.e., Zermelo--Fraenkel set theory without the axiom of choice), avoiding any metamathematical considerations.