The topological pigeonhole principle for ordinals

Jacob Hilton

Published 2014-10-09Version 1

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