Some remarks on uncountable rainbow Ramsey theory

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.