On constructions with 22-cardinals

We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman's neat simplified morasses called $2$-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces. A new result which we obtain as a side product is the consistency of the existence of a function $f:[\lambda^{++}]^2\rightarrow[\lambda^{++}]^{\leq\lambda}$ with the appropriate $\lambda^+$-version of property $\Delta$ for regular $\lambda\geq\omega$ satisfying $\lambda^{<\lambda}=\lambda$.Comment: Minor correction

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