On the arithmetic of density

The κ  -density of a cardinal μ≥κμ≥κ is the least cardinality of a dense collection of κ-subsets of μ   and is denoted by D(μ,κ)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≤μλ≤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 Theorem 2.6. If  κ=cfκ<θ=cfμ<μand the set of cardinals  λ<μλ<μof 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 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.