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Currently displaying 1 – 4 of 4

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Continuous tree-like scales

James Cummings — 2010

Open Mathematics

Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at ℵω is consistent with Martin’s Maximum.

More results in polychromatic Ramsey theory

Uri Abraham; James Cummings — 2012

Open Mathematics

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 $ω_{2} ↛^{p o l y} {(ω_{1})}_{2 - b d d}^{2}$ .

Singular cardinals and strong extenders

Arthur Apter; James Cummings; Joel Hamkins — 2013

Open Mathematics

We investigate the circumstances under which there exist a singular cardinal µ and a short (κ,µ)-extender E witnessing “κ is µ-strong”, such that µ is singular in Ult(V, E).

A consistency result on weak reflection

James Cummings; Mirna Džamonja; Saharon Shelah — 1995

Fundamenta Mathematicae

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