  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 of uncountable cofinality, while its successor 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 for which there is a
collection of graphs on $\kappa^+$ whose size is less than $2^\kappa$ and such
that any graph on $\kappa^+$ embeds into one of the graphs in the collection.
