  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.
