Ramsey theory for monochromatically well-connected subsets

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 $\to$.