On the arithmetic of density

The κ-density of a cardinal μ≥κ is the least cardinality of a dense collection of κ-subsets of μ and is denoted by D(μ,κ). The Singular Density Hypothesis (SDH) for a singular cardinal μ of cofinality cfμ=κ is the equation D‾(μ,κ)=μ⁺, where D‾(μ,κ) is the density of all unbounded subsets of μ of ordertype κ. The Generalized Density Hypothesis (GDH) for μ and λ such that λ≤μ is:D(μ,λ)={μ if cfμ≠cfλμ⁺ if cfμ=cfλ. Density is shown to satisfy Silver's theorem. The most important case is: Theorem 2.6 If κ=cfκ<θ=cfμ<μ and the set of cardinals λ<μ of cofinality κ that satisfy the SDH is stationary in μ then the SDH holds at μ. A more general version is given in Theorem 2.8. A corollary of Theorem 2.6 is: Theorem ⊗ If the Singular Density Hypothesis holds for all sufficiently large singular cardinals of some fixed cofinality κ, then for all cardinals λ with cfλ≥k, for all sufficiently large μ, the GDH holds.