  We define well-connectedness, an order-theoretic notion of largeness whose
associated partition relations $\nu\to_{wc}(\mu)_\lambda^2$ formally weaken
those of the classical Ramsey relations $\nu\to(\mu)_\lambda^2$. We show that
it is consistent that the arrows $\to_{wc}$ and $\to$ are, in infinite
contexts, essentially indistinguishable. We then show, in contrast, that in
Mitchell's model of the tree property at $\omega_2$, the relation
$\omega_2\to_{wc}(\omega_2)_\omega^2$ does hold, and that the consistency
strength of this relation holding is precisely a weakly compact cardinal. These
investigations may be viewed as augmenting those of [BHS], the central arrow of
which, $\to_{hc}$, is of intermediate strength between $\to_{wc}$ and the
Ramsey arrow $\to$.
