On the indecomposability of $ω^n$

The topological pigeonhole principle for ordinals

Given a cardinal $κ$ and a sequence $\left(α_i\right)_{i\inκ}$ of ordinals, we determine the least ordinal $β$ (when one exists) such that the topological partition relation \[β\rightarrow\left(top\,α_i\right)^1_{i\inκ}\] holds, including an independence result for one class of cases. Here the prefix "$top$" means that the homogeneous set must have the correct topology rather than the correct order type. The answer is linked to the non-topological pigeonhole principle of Milner and Rado.