Mycielski ideals and uniform trees
Volume 264 / 2024
Abstract
Mycielski introduced ideals $\mathfrak {P}_{k}$ and $\mathfrak {C}_{k}$ on ${}^{\omega}{k}$ which are called Mycielski ideals. We investigate their cardinal invariants. Strengthening results of Kamo and of Ros{ł}anowski we prove that $\mathrm {cof}(\mathfrak {P}_{k})$ and $\mathrm {cof}(\mathfrak {C}_{k})$ may be singular and that their cofinality is $ \gt 2^{\aleph _0}$. We study the relation between $\mathfrak {P}_{k}$ and the ideal associated with Silver forcing and $\mathfrak {C}_{k}$ and the ideal associated with uniform Sacks forcing. We prove that consistently neither ideal is contained in one of the others. We show that $\mathfrak {P}_{k}$ and $\mathfrak {C}_{k}$ are Tukey reducible to $\mathfrak {P}_{k+1}$ and $\mathfrak {C}_{k+1}$, respectively. By iterating a uniform version $\mathbb {U}_k$ of $k$-dimensional Sacks forcing we produce a model in which $\mathrm {cov}(\mathfrak {C}_{k+1}) = \aleph _1$ and $\mathrm {cov}(\mathfrak {C}_{k}) = \aleph _2$. We prove that the natural amoeba $\mathbb {A}(\mathbb {U}_{k})$ of $\mathbb {U}_k$ has the Laver property, and thus, when iterated, increases the additivity of the ideal associated with $\mathbb {U}_k$ without adding Cohen or random reals.
