A framework for forcing constructions at successors of singular cardinals

Cummings, James; Džamonja, Mirna; Magidor, Menachem; Morgan, Charles; Shelah, Saharon

doi:10.1090/tran/6974

A framework for forcing constructions at successors of singular cardinals
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by James Cummings, Mirna Džamonja, Menachem Magidor, Charles Morgan and Saharon Shelah PDF

Trans. Amer. Math. Soc. 369 (2017), 7405-7441 Request permission

Abstract:

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal $\kappa$ of uncountable cofinality, while $\kappa ^+$ enjoys various combinatorial properties.

As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $\kappa$ of uncountable cofinality where SCH fails and such that there is a collection of size less than $2^{\kappa ^+}$ of graphs on $\kappa ^+$ such that any graph on $\kappa ^+$ embeds into one of the graphs in the collection.

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