Ignore that last post…

Actually, I think I lied to you in the last post, at least as far as our strategy is concerned. I’ll outline another route to the result that is closer to what Shelah does; it’s actually shorter than what I had in mind originally.

Recalling our situation, ${\mu}$ is a singular cardinal, and ${\mathfrak{A}=\langle H(\chi), \in, <_\chi\rangle}$ is as usual. We have also assumed ${\aleph_0<{\rm cf}(\sigma)\leq\sigma\leq{\rm cf}(\mu)<\theta\leq\kappa<\mu}$. Let us assume that ${\mathcal{F}}$ is a family of functions as required in the definition of ${\lambda_2}$, i.e., a collection of functions that are “${\sigma}$-cofinal in any subproduct of size ${<\theta}$ constructed from ${\prod({\sf Reg}\cap(\kappa,\mu)}$” (see previous posts for a more precise definition, but things should be clear from the construction).

Given a finite subset ${F}$ of ${\mathcal{F}}$, we define

$\displaystyle A_F:=\mu\cap{\rm Sk}_{\mathfrak{A}}(\kappa+1\cup\{\mu\}\cup F), \ \ \ \ \ (1)$

and define

$\displaystyle \mathcal{P}=\{A_F:F\in [\mathcal{F}]^{<\aleph_0}\}. \ \ \ \ \ (2)$

Note that each ${A_F}$ is a subset of ${\mu}$ of cardinality ${\kappa}$, so

$\displaystyle \mathcal{P}\subseteq [\mu]^\kappa. \ \ \ \ \ (3)$

Also, it should be clear that

$\displaystyle |\mathcal{P}|\leq |[\mathcal{F}]^{<\aleph_0}|=|\mathcal{F}|, \ \ \ \ \ (4)$

so we will be finished if we can prove that any ${Y\in [\mu]^{<\theta}}$ can be covered by a union of fewer than ${\sigma}$ sets drawn from ${\mathcal{P}}$.

How will we do this? Given ${Y\in [\mu]^{<\theta}}$, we will produce a family ${H\subseteq \mathcal{F}}$ satisfying the following two properties:

$\displaystyle |H|<\sigma, \ \ \ \ \ (5)$

and

$\displaystyle Y\subseteq\mu\cap{\rm Sk}_\mathfrak{A}(\kappa+1\cup\{\mu\}\cup H). \ \ \ \ \ (6)$

This suffices, because

$\displaystyle \mu\cap{\rm Sk}_\mathfrak{A}(\kappa+1\cup\{\mu\}\cup H)=\bigcup\{A_F: F\in [H]^{<\aleph_0}\}, \ \ \ \ \ (7)$

and since ${|[H]^{<\aleph_0}|<\sigma}$, we will have shown that ${Y}$ can be covered by a union of fewer than ${\sigma}$ sets from ${\mathcal{P}}$.