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

Abstract

We present a number of results concerning infinite-exponent partition relations on linear orders of the form $\langle {}^\alpha 2,<_{\text{lex}}\rangle$ 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 $\langle {}^\alpha 2,<_{\text{lex}}\rangle \rightarrow (\tau)^\tau$ for $\tau$ countable.