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.