A framework for forcing constructions at successors of singular cardinals

Abstract. We describe a framework for proving consistency re-sults about singular cardinals of arbitrary cofinality and their suc-cessors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal κ of uncountable cofinality, while κ+ enjoys various combinatorial prop-erties. 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 of uncountable cofinality where SCH fails and such that there is a collection of size less than 2κ of graphs on κ+ such that any graph on κ+ embeds into one of the graphs in the collection