On the arithmetic of density

Abstract

The $κ$-density of a cardinal $μ\geκ$ is the least cardinality of a dense collection of $κ$-subsets of $μ$ and is denoted by $\mathcal D(μ,κ)$. The Singular Density Hypothesis (SDH) for a singular cardinal $μ$ of cofinality $cfμ=κ$ is the equation $\mathcal D(μ,κ)=μ^+$. The Generalized Density Hypothesis (GDH) for $μ$ and $λ$ such that $λ\leμ$ is: $\mathcal D(μ,λ)=μ$ if $cfμ\not=cfλ$ and $\mathcal D(μ,λ)=μ^+$ 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 \textsf{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 3.2 If the Singular Density Hypothesis holds for all sufficiently large singular cardinals of some fixed cofinality $κ$, then for all cardinals $λ$ with $cfλ\ge κ$, for all sufficiently large $μ$, the GDH holds.