Ramsey theory for monochromatically well-connected subsets

Abstract

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 $\to$.