## [Sh:430.1-4] An easy inequality

January 31, 2011 at 16:38 | Posted in Cardinal Arithmetic, cov vs. pp, [Sh:355], [Sh:430] | 1 Comment

What follows is not really part of the first section of [Sh:430], but it does pin down an important connection between cov and pp. The proposition is a special case of part of Shelah’s “cov vs. pp theorem”, Theorem 5.4 on page 87 of Cardinal Arithmetic. Note that there are no special assumptions on ${\mu}$ here. Getting inequalities in the reverse direction (i.e., showing that covering numbers are less than pp numbers) is generally a more difficult proposition.

Let us recall that ${{\rm cov}(\mu,\mu,\kappa^+, 2)}$ is the minimum cardinality of a set ${\mathcal{P}\subseteq[\mu]^{<\mu}}$ such that for any ${A\in [\mu]^\kappa}$ (equivalently ${[\mu]^{\leq\kappa}}$), there is a ${B\in\mathcal{P}}$ such that ${A\subseteq B}$.

Proposition 1 If ${\mu}$ is singular of cofinality ${\kappa}$, then

$\displaystyle {\rm pp}(\mu)\leq {\rm cov}(\mu,\mu,\kappa^+,2). \ \ \ \ \ (1)$

Proof: Suppose by way of contradiction that

$\displaystyle \theta:={\rm cov}(\mu,\mu,\kappa^+,2)<{\rm pp}(\mu). \ \ \ \ \ (2)$

By the definition of ${\rm pp}$, we can find a (not necessarily increasing) sequence of regular cardinals ${\langle \mu_i:i<\kappa\rangle}$ and an ultrafilter ${D}$ on ${\kappa}$ such that

$\displaystyle \{i<\kappa:\mu_i\leq\tau\}\notin D\text{ for each }\tau<\mu, \ \ \ \ \ (3)$

and

$\displaystyle \theta<\sigma:={\rm cf}\left(\prod_{i<\kappa}\mu_i, <_D\right). \ \ \ \ \ (4)$

Let ${\mathcal{P}\subseteq [\mu]^{<\mu}}$ be a family of cardinality ${\theta}$ standing as witness to ${{\rm cov}(\mu,\mu,\kappa^+,2)=\theta}$, and let ${\langle f_\alpha:\alpha<\sigma\rangle}$ be a ${<_D}$-increasing and cofinal sequence of functions in ${\prod_{i<\kappa}\mu_i}$. For each ${\alpha<\sigma}$, the range of ${f_\alpha}$ is a subset of ${\mu}$ of cardinality at most ${\kappa}$, and so we can find ${A_\alpha\in \mathcal{P}}$ such that

$\displaystyle {\rm ran}(f_\alpha)\subseteq A_\alpha. \ \ \ \ \ (5)$

Since ${\sigma}$ is a regular cardinal greater than ${\theta}$, there is a single ${A^*\in \mathcal{P}}$ such that

$\displaystyle |\{\alpha<\sigma: A_\alpha=A^*\}|=\sigma. \ \ \ \ \ (6)$

Thus, (by passing to a subsequence of ${\langle f_\alpha:\alpha<\sigma\rangle}$) we may as well assume that the range of each ${f_\alpha}$ is a subset of ${A^*}$.

But ${|A^*|<\mu}$, and so

$\displaystyle B:= \{i<\kappa:|A^*|<\mu_i\}\in D \ \ \ \ \ (7)$

by our choice of ${D}$. Let us now define a function ${g}$ by setting ${g(i)=0}$ if ${i\notin B}$, and

$\displaystyle g(i)=\sup(A^*\cap\mu_i) \ \ \ \ \ (8)$

whenever ${i}$ is in ${B}$. Our assumptions imply that ${g}$ is in ${\prod_{i<\kappa}\mu_i}$.

If ${\alpha<\sigma}$, then for each ${i\in B}$ we have

$\displaystyle f_\alpha(i)\leq g(i) \ \ \ \ \ (9)$

as ${f_\alpha(i)\in A^*\cap\mu_i}$.

It follows that ${g}$ is a function in ${\prod_{i<\kappa}\mu_i}$ such that${f_\alpha\leq_D g}$ for all ${\alpha<\sigma}$. This is absurd, given our choice of ${\langle f_\alpha:\alpha<\sigma\rangle}$, and the proof is complete. $\Box$

[Updated 2-7-11]

## [Sh:430.1-2] Framework for proof

Let us assume that ${\mu}$ is a singular cardinal with ${\rm{pp}(\mu)=\mu^+}$. We will prove that there is a family ${\mathcal{P}\subseteq [\mu]^{<\mu}}$ of cardinality ${\mu^+}$ such that every member of ${[\mu]^{\rm{cf}(\mu)}}$ is a subset of some member of ${\mathcal{P}}$.

Fix a sufficiently large regular cardinal ${\chi}$; we will be working with elementary submodels of the structure ${\langle H(\chi),\in, <_\chi\rangle}$ where ${<_\chi}$ is some appropriate well-ordering of ${H(\chi)}$ used to give us definable Skolem functions.

Let ${\mathfrak{M}=\langle M_\alpha:\alpha<\mu^+\rangle}$ be a ${\mu^+}$-approximating sequence, that is, ${\mathfrak{M}}$ is a ${\in}$-increasing and continuous sequence of elementary submodels of ${H(\chi)}$ such that for each ${\alpha<\mu^+}$,

• ${\mu\in M_0}$,
• ${M_\alpha}$ has cardinality ${\mu}$,
• ${M_\alpha\cap\mu^+}$ is an initial segment of ${\mu^+}$ (so ${\mu+1\subseteq M_\alpha\cap\mu^+}$), and
• ${\langle M_\beta:\beta<\alpha\rangle\in M_{\alpha+1}}$.

Let ${M^*}$ denote ${\bigcup_{\alpha<\mu^+} M_\alpha}$. Then ${M^*}$ is an elementary submodel of ${H(\chi)}$ of cardinality ${\mu^+}$ containing ${\mu^+}$ as both an element and subset. Define

$\displaystyle \mathcal{P}=M^*\cap [\mu]^{<\mu}. \ \ \ \ \ (1)$

Clearly ${\mathcal{P}\subseteq [\mu]^{<\mu}}$ and ${|\mathcal{P}|=\mu^+}$, so we need only verify that for any ${a\in [\mu]^{\rm{cf}(\mu)}}$, there is a ${b\in \mathcal{P}}$ with ${a\subseteq b}$.

In broad terms, this will be done via an “${I[\lambda]}$ argument” with ${\lambda=\mu^+}$, but we’ll fill in the details in further posts.

## [Sh:430.1-1] What are we aiming for?

We focus our attention on the first section of [Sh:430]: “equivalence of two covering properties”. The first result states the following:

Theorem 1 If ${\rm{pp}(\lambda)=\lambda^+}$, ${\lambda>\rm{cf}(\lambda)=\kappa>\aleph_0}$ then ${\rm{cov}(\lambda,\lambda,\kappa^+,2)=\lambda^+}$.

I am pretty sure that this isn’t what Shelah meant to write here — the assumption that ${\lambda}$ has uncountable cofinality means that the result is a trivial consequence of a deeper theorem in Cardinal Arithmetic [Update: Perhaps not! See 2nd update below]. I think that the proof he gives works even for the countable cofinality case, and this gives us something of interest because ${\lambda}$ could very well be a fixed point. So, here’s what I conjecture he meant to say (after changing notation and translating the “cov” statement into more standard form):

Theorem 2 Let ${\mu}$ be a singular cardinal. Then ${\rm{pp}(\mu)=\mu^+}$ if and only if there is a family ${\mathcal{P}\subseteq [\mu]^{<\mu}}$ of cardinality ${\mu^+}$ such that every member of ${[\mu]^{\rm{cf}(\mu)}}$ is a subset of some element of ${\mathcal{P}}$.

We’ll work through his proof as best we can, and see if my conjecture is correct.

UPDATE 1: I don’t think the conjecture is correct. It looks to me like the proof was originally written for singular of countable cofinality, but a mistake was discovered and the statement was corrected but a lot of the old proof didn’t get revised properly. The annotated content was revised, but the abstract was not. Anyway, I’ll ask Saharon if I can’t figure out what’s going on.

UPDATE 2: I think the theorem doesn’t follow from the Cardinal Arithmetic stuff. The problem I run into is that when Shelah says informally that “cov = pp when the cardinal has uncountable cofinality”, there are lots of disclaimers hidden in the background. But this underscores why I think what I’m doing here is important — I want to pin down what exactly is known and what exactly is still open in this area.