The uncountable spectra of countable theories.

Hart, Bradd; Hrushovski, Ehud; Laskowski, Michael C.

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We study polychromatic Ramsey theory with a focus on colourings of [ω 2]2. We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2ω = ω 2 and $ω_{2} \to^{p o l y} {(α)}_{ℵ_{0} - b d d}^{2}$ for every α <ω 2; (2) 2ω = ω 2 and $ω_{2} ↛^{p o l y} {(ω_{1})}_{2 - b d d}^{2}$ .

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Another proof of a result of Jech and Shelah

Péter Komjáth (2013)

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Shelah’s pcf theory describes a certain structure which must exist if $ℵ_{ω}$ is strong limit and $2^{ℵ_{ω}} > ℵ_{ω_{1}}$ holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that...