Infinite-Exponent Partition Relations on Higher Analogues of the Real Line

We present a number of results concerning infinite-exponent partition relations on linear orders of the form $⟨{}^{\alpha }2,{<}_{\text{lex}}⟩$ for $\alpha$ an ordinal, generalising the setting of the real line, working throughout in ZF without the Axiom of Choice. As a particular consequence of our results, we obtain a full classification of the relation $⟨{}^{\alpha }2,{<}_{\text{lex}}⟩\to (\tau {)}^{\tau }$ for $\tau$ countable.