  An infinite graph is said to be highly connected if the induced subgraph on
the complement of any set of vertices of smaller size is connected. We continue
the study of weaker versions of Ramsey Theorem on uncountable cardinals
asserting that if we color edges of the complete graph we can find a large
highly connected monochromatic subgraph. In particular, several questions of
Bergfalk, Hru\v{s}\'ak and Shelah are answered by showing that assuming the
consistency of suitable large cardinals the following are relatively consistent
with $\mathsf{ZFC}$: $\kappa\to_{hc} (\kappa)^2_\omega$ for every regular
cardinal $\kappa\geq \aleph_2$ and $\neg\mathsf{CH}+ \aleph_2 \to_{hc}
(\aleph_1)^2_\omega$. Building on a work of Lambie-Hanson, we also show that
$\aleph_2 \to_{hc} [\aleph_2]^2_{\omega,2}$ is consistent with
$\neg\mathsf{CH}$. To prove these results, we use the existence of ideals with
strong combinatorial properties after collapsing suitable large cardinals.
