On the arithmetic of density

The $\kappa$-density of a cardinal $\mu \ge \kappa$ is the least cardinality of a dense collection of $\kappa$-subsets of $\mu$ and is denoted by $\mathcal D(\mu,\kappa)$\mathrm{.}TheSingularDensityHypothesis(SDH)forasingularcardinal$\mu$ofcofinality$cf\mu=\kappa$istheequation$\mathcal D(\mu,\kappa)=\mu^+$\mathrm{.}TheGeneralizedDensityHypothesis(GDH)for$\mu$and$\lambda$suchthat$\lambda\le\mu$is:$\mathcal D(\mu,\lambda)=\mu$if$cf\mu\not=cf\lambda$and$\mathcal D(\mu,\lambda)=\mu^+$if$cf\mu=cf\lambda$. Density is shown to satisfy Silver’s theorem. The most important case is: Theorem 2.6. If $\kappa =cf\kappa <\theta =cf\mu <\mu$ and the set of cardinals $\lambda <\mu$ of cofinality $\kappa$ that satisfy the $\mathsf{\text{SDH}}$ is stationary in $\mu$ then the SDH holds at $\mu$. A more general version is given in Theorem 2.8 A corollary of Theorem 2.6 is: Theorem 3.2 If the Singular Density Hypothesis holds for all sufficiently large singular cardinals of some fixed cofinality $\kappa$, then for all cardinals $\lambda$ with $cf\lambda \ge \kappa$, for all sufficiently large $\mu$, the GDH holds.