Towers and gaps at uncountable cardinals

Abstract

Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either p(κ) = t(κ) or there is a (p(κ), λ)-gap of club-supported slaloms for some λ < p(κ). While the existence of such gaps is unclear, this is a promising step to lift Malliaris and Shelah's proof of p = t to uncountable cardinals. We do analyze gaps of slaloms and, in particular, show that p(κ) is always regular; the latter extends results of Garti. Finally, we turn to club variants of p(κ) and present a new model for the inequality p(κ) = κ + < p cl (κ) = 2 κ . In contrast to earlier arguments by Shelah and Spasojević, we achieve this by adding κ-Cohen reals and then successively diagonalizing the club filter; the latter is shown to preserve a Cohen witness to p(κ) = κ + .

Strongly unfoldable, splitting and bounding

“…A proof of $$\mathfrak {p}_{cl}(\kappa ) = \mathfrak {b}(\kappa )$$ for a regular uncountable κ is given in [6, Observation 4.2]. A consistency proof of $$\mathfrak {p}(\kappa ) = \kappa ^+ &lt; \mathfrak {b}(\kappa )$$, which uses the identity $$\mathfrak {p}_{cl}(\kappa ) = \mathfrak {b}(\kappa )$$, is given in [6, § 4]. The relevant poset for this proof and for our proof in § 4 is the following: Definition Let $$\mathcal {C}$$ denote the collection of all clubs in κ.…”

“…So the poset that we are forcing with is $$\mathbb {P} = \mathbb {C}_{\kappa ^+}* \dot{\mathbb {M}}(\mathcal {C})_\lambda$$, where $$\dot{\mathbb {M}}(\mathcal {C})_\lambda$$ is a $$\mathbb {C}_{\kappa ^+}$$‐name for the $$&lt;\kappa$$‐support iteration of $$\mathbb {M}(\mathcal {C})$$ (from Definition 2.6) of length λ. The following property and notation have been established in [6]: 1.This forcing $$\mathbb {P}$$ has the $$\kappa ^+$$‐c.c., is κ‐closed and forces that $$\mathfrak {c}(\kappa ) = \lambda$$. …”

The structure of $$\kappa $$-maximal cofinitary groups

“…The significance of this definition is that the forcing axiom for the class of <κclosed, κ-centered posets with canonical lower bounds is equivalent to p(κ) = 2 κ . More precisely, Theorem 3.3 [Theorem 1.8 of [3]] If P is a <κ-closed, κ-centered poset with canonical lower bounds and below every p ∈ P there is a κ-sized antichain. Then for any collection D of <p(κ) many dense subsets of P, there is a filter G ⊆ P which meets every element of D.…”

“…It suffices to ensure that for each U γ and each word w ∈ W α the permutation w(h) U γ : U γ → U γ has <κ many fixed points. 3 We will construct h U γ differently depending on f α .…”

“…Specifically he proves that under MA, for any n < ω and any partition {O k | k < n} of ω, there is a maximal cofinitary group G whose orbits are exactly this partition. Building on the above mentioned generalization of Bell's theorem (see [10], as well as [3]), we establish the following higher Baire spaces analogue: Theorem [See Theorem 3.9] If p(κ) = 2 κ then given any λ < κ and any partition {O α | α < λ} of κ there is a κ-maximal cofinitary group G so that the orbits of G are exactly {O α | α < λ}.…”