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Ramsey theory for monochromatically well-connected subsets

Volume 249 / 2020

Jeffrey Bergfalk Fundamenta Mathematicae 249 (2020), 95-103 MSC: 03E02, 03E55. DOI: 10.4064/fm713-7-2019 Published online: 28 October 2019


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 $.


  • Jeffrey BergfalkCentro de Ciencias Matemáticas
    A.P. 61-3, Xangari
    Morelia, Michoacán, 58089, México

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