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Journal Article

Universal Graphs at the Successor of a Singular Cardinal

Mirna Džamonja and Saharon Shelah
The Journal of Symbolic Logic
Vol. 68, No. 2 (Jun., 2003), pp. 366-388
Stable URL: https://www.jstor.org/stable/4147689
Page Count: 23

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Abstract

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are $\mu^{++}$ graphs on μ+ that taken jointly are universal for the graphs on μ+, while $2^{\mu^+} \gg \mu^{++}$. The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.

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