Abstract
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.
Highlights
The class of uncountable regular cardinals is naturally divided into three disjoint classes: the successors of regular cardinals, the successors of singular cardinals and the weakly inaccessible cardinals
When we consider a combinatorial question about uncountable regular cardinals, typically these classes require separate treatment and very frequently the successors of singular cardinals present the hardest problems
To give some context for our work, we review a standard strategy for proving consistency results about the successors of regular cardinals
Summary
The class of uncountable regular cardinals is naturally divided into three disjoint classes: the successors of regular cardinals, the successors of singular cardinals and the weakly inaccessible cardinals. Λ κ Cohen forcing to add many subsets of a regular cardinal , which we denote by Add(κ, λ), is the collection of partial functions of size less than κ from λ to 2, Fn(λ, 2, κ), ordered by reverse inclusion. The ordering is that p = (tp, f p) ≤ q = (tq, f q) t t if q is obtained by, in Kunen's [15] vivid précis, `sawing o p parallel to the ground': i.e. there is some α < κ+ such that tq = tp ∩ α2, dom(f q) ⊆ dom(f p) and f q(ξ)
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