On the arithmetic of density

Abstract

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 $\mathcal{D}(\mu ,\kappa )$. The Singular Density Hypothesis (SDH) for a singular cardinal $\mu$ of cofinality $cf\mu =\kappa$ is the equation $\mathcal{D}(\mu ,\kappa )={\mu }^{+}$. The Generalized Density Hypothesis (GDH) for $\mu$ and $\lambda$ such that $\lambda \le \mu$ is: $\mathcal{D}(\mu ,\lambda )=\mu$ if $cf\mu \ne cf\lambda$ 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 \textsf{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.