Ramsey theory for monochromatically well-connected subsets

We define well-connectedness, an order-theoretic notion of largeness whose associated partition relations v ->(wc) (mu)(lambda)(2) formally weaken those of the classical Ramsey relations v -> (mu)(lambda)(2). We show that it is consistent that the arrows ->(wc) and -> 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) -> (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 Bergfalk et al. (2018), the central arrow of which, ->(hc) is of intermediate strength between ->(wc) and the Ramsey arrow ->.