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 $cf\mu\not=cf\lambda$ 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 \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.