A framework for forcing constructions at successors of singular cardinals

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 κ \kappa of uncountable cofinality, while κ + \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 κ \kappa of uncountable cofinality where SCH fails and such that there is a collection of size less than 2 κ + 2^{\kappa ^+} of graphs on κ + \kappa ^+ such that any graph on κ + \kappa ^+ embeds into one of the graphs in the collection.