Ramsey theory for monochromatically well-connected subsets
Volume 249 / 2020
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 Bergfalk et al. (2018), the central arrow of which, $\to _{hc}$, is of intermediate strength between $\to _{wc}$ and the Ramsey arrow $\to $.
