A framework for forcing constructions at successors of singular cardinals

by Cummings and Dzamonja and Magidor and Morgan and Shelah. [CDMMSh:963]

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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