The topological pigeonhole principle for ordinals

TL;DR

This paper establishes the minimal ordinal for topological partition relations involving sequences of ordinals, extending classical pigeonhole principles to topological contexts and providing independence results.

Contribution

It introduces the topological pigeonhole principle for ordinals, determining the least ordinal satisfying specific topological partition relations and connecting it to known non-topological principles.

Findings

Determined the least ordinal for topological partition relations.

Established an independence result for certain cases.

Linked topological and non-topological pigeonhole principles.

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 \[\beta\rightarrow\left(top\,\alpha_i\right)^1_{i\in\kappa}\] holds, including an independence result for one class of cases. Here the prefix "$top$" means that the homogeneous set must have the correct topology rather than the correct order type. The answer is linked to the non-topological pigeonhole principle of Milner and Rado.