%0 Journal Article %A Cummings, James %A Dzamonja, Mirna %A Magidor, Menachim %A Morgan, Charles %A Shelah, Saharon %D 2014 %T A framework for forcing constructions at successors of singular cardinals %U https://kilthub.cmu.edu/articles/journal_contribution/A_framework_for_forcing_constructions_at_successors_of_singular_cardinals/6476519 %R 10.1184/R1/6476519.v1 %2 https://kilthub.cmu.edu/ndownloader/files/11907959 %K Mathematical Sciences %X

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.

%I Carnegie Mellon University