Some remarks on uncountable rainbow Ramsey theory

TL;DR

This paper explores advanced rainbow Ramsey theorems at large cardinals, revealing new independence results, constructions, and questions about their behavior under various set-theoretic assumptions and forcings.

Contribution

It demonstrates that certain rainbow Ramsey properties do not characterize large cardinals, provides simplified models, and investigates their indestructibility under forcing.

Findings

Rainbow Ramsey properties do not characterize weak compactness.

Constructed a model where a specific rainbow partition relation fails.

Showed indestructibility of certain colorings under strongly proper forcing.

Abstract

We discuss the rainbow Ramsey theorems at limit cardinals and successors of singular cardinals, addressing some questions in \cite{MR2354904} and \cite{MR2902230}. In particular, we show for inaccessible $\kappa$, $\kappa\to^{poly}(\kappa)^2_{2-bdd}$ does not characterize weak compactness and for singular $\kappa$, $\mathrm{GCH}+\square_\kappa$ implies $\kappa^+\not\to^{poly} (\eta)^2_{<\kappa-bdd}$ for any $\eta\geq cf(\kappa)^+$ and $\kappa^+\to^{poly} (\nu)^2_{<\kappa-bdd}$ for any $\nu<cf(\kappa)^+$. We also provide a simplified construction of a model for $\omega_2\not\to^{poly} (\omega_1)^2_{2-bdd}$ originally constructed in \cite{MR2902230} and show the witnessing coloring is indestructible under strongly proper forcings but destructible under some c.c.c forcing. Finally, we conclude with some remarks and questions on possible generalizations to rainbow partition relations for triples.