More Ramsey theory for highly connected monochromatic subgraphs

Abstract

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.