The topological pigeonhole principle for ordinals

Abstract

Given a cardinal $\kappa$ and a sequence ${\left({\alpha }_i\right)}_{i\in \kappa }$ of ordinals, we determine the least ordinal $\beta$ (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.

arXiv comment24 pages

Cited by in corpus (3)

• The Polarised Partition Relation for Order Types
• Calculating the closed ordinal Ramsey number $R^{cl}(\omega \cdot 2,3{)}^2$
• The closed ordinal Ramsey number $R^{cl}({\omega }^2,3)={\omega }^6$