Stationary and Closed Rainbow subsets

TL;DR

This paper explores the structured rainbow Ramsey theory at uncountable cardinals, focusing on stationary and closed rainbow subsets, and uncovers connections with Chang's Conjectures and partition relations.

Contribution

It introduces the study of stationary and closed rainbow subsets at uncountable cardinals and links these concepts to Chang's Conjectures, addressing open questions.

Findings

Established connections between rainbow Ramsey properties and Chang's Conjectures.

Extended rainbow Ramsey theory to include stationary and closed subsets.

Provided new insights into partition relations at uncountable cardinals.

Abstract

We study the structured rainbow Ramsey theory at uncountable cardinals. When compared to the usual rainbow Ramsey theory, the variation focuses on finding a rainbow subset that not only is of a certain cardinality but also satisfies certain structural constraints, such as being stationary or closed in its supremum. In the process of dealing with cardinals greater than $\omega_1$, we uncover some connections between versions of Chang's Conjectures and instances of rainbow Ramsey partition relations, addressing a question raised in \cite{zhang}.