\DeclareMathOperator{\pp}{pp}

\def\pcf{\rm{pcf}}

\DeclareMathOperator{\cov}{cov}

\def\cf{\rm{cf}}

\def\REG{\sf {REG}}

\def\restr{\upharpoonright}

\def\bd{\rm{bd}}

\def\subs{\subseteq}

\def\cof{\rm{cof}}

\def\ran{\rm{ran}}

\DeclareMathOperator{\PP}{pp}

\)

This is a direct continuation of the previous post. I want to refresh your memory about some definitions, and then proceed to a proof of the easy parts of the chain of inequalities we were investigating.

We have assumed $\cf(\lambda)<\theta=\cf(\theta)<\lambda$.

**Definition**

- $\pp_{<\theta}(\lambda)$ is defined to be the supremum of all cardinals of the form $\cof(\prod\mathfrak{a}/\mathcal{U})$, where
- $\mathfrak{a}$ is cofinal in $\lambda\cap\REG$,
- $|\mathfrak{a}|<\theta$, and
- $\mathcal{U}$ is an ultrafilter on $\mathfrak{a}$ disjoint to the ideal of bounded subsets of $\mathfrak{a}$.
- $\cf_{<\theta}(\prod(\lambda\cap\REG), <_{J^\bd_\lambda})$ is the minimum cardinality of a family of functions $\mathcal{F}\subseteq\prod(\lambda\cap\REG)$ such that for any cofinal $\mathfrak{a}\subseteq\lambda\cap\REG$ of cardinality $\lambda$ and any $g\in\prod\mathfrak{a}$, there is an $f\in \mathcal{F}$ such that $g<f\restr\mathfrak{a}$ modulo the bounded ideal on $\mathfrak{a}$
- $\cov(\lambda,\lambda,\theta,2)$ is the minimum cardinality of a family $\mathcal{P}\subseteq [\lambda]^{<\lambda}$ such that for any $B\in [\lambda]^{<\theta}$ there is a $A\in\mathcal{P}$ with $B\subseteq A$

We will show

$\pp_{<\theta}(\lambda)\leq\cf_{<\theta}(\prod(\lambda\cap\REG), <_{J^\bd_\lambda})\leq\cov(\lambda,\lambda,\theta,2).$

This gives us the first two inequalities in our chain, and both are easy.

For the first, suppose $\mathcal{F}$ is as described, and let $\mathfrak{a}$ and $\mathcal{U}$ be as in the definition of $\pp_{<\theta}(\lambda)$

The set $$\{\mathcal{f}\restr\mathfrak{a}:f\in\mathcal{F}\}$$

(or rather the set of equivalence classes of these functions modulo $\mathcal{U}$) is cofinal in $\prod\mathfrak{a}/\mathcal{U}$ because $U$ is disjoint to the bounded ideal on $\mathfrak{a}$ and the inequality follows immediately.

For the second, suppose $\mathcal{P}\subseteq[\lambda]^{<\lambda}$ is as described above.

For $A\in\mathcal{P}$, let

us define a function $f_A\in\prod(\lambda\cap \REG$) by

$$f_A(\kappa)=

\begin{cases}

\sup(A\cap \kappa)+1 & \text{if this is less than $\kappa$,}\\

0 &\text{otherwise}.

\end{cases}

$$

Does this collection of functions work? Suppose $B$ is a cofinal subset of $\lambda\cap\REG$ of cardinality $<\theta$, and let $g\in\prod B$. Choose $A\in\mathcal{P}$ such that $\ran(g)\subseteq A$.

If $\kappa\in B$ is sufficiently large (say $|A|<\kappa$), then

$$g(\kappa)\leq\sup(A\cap\kappa)<f_A(\kappa)<\kappa,$$

and we are done.