  We study a pair of weakenings of the classical partition relation $\nu
\rightarrow (\mu)^2_\lambda$ recently introduced by
Bergfalk-Hru\v{s}\'{a}k-Shelah and Bergfalk, respectively. Given an
edge-coloring of the complete graph on $\nu$-many vertices, these weakenings
assert the existence of monochromatic subgraphs exhibiting high degrees of
connectedness rather than the existence of complete monochromatic subgraphs
asserted by the classical relations. As a result, versions of these weakenings
can consistently hold at accessible cardinals where their classical analogues
would necessarily fail. We prove some complementary positive and negative
results indicating the effect of large cardinals, forcing axioms, and square
principles on these partition relations. We also prove a consistency result
indicating that a non-trivial instance of the stronger of these two partition
relations can hold at the continuum.
