## Strategy and assumptions

In this post, we’ll lay out the notation and strategy for the proof.

Let us assume ${\sigma}$, ${\theta}$, ${\mu}$, and ${\lambda}$ are cardinals with

$\displaystyle \aleph_0<\sigma\leq{\rm cf}(\mu)<\theta\leq\mu, \ \ \ \ \ (1)$

such that

$\displaystyle {\rm tcf}(\prod\mathfrak{a}, <_I)<\lambda \ \ \ \ \ (2)$

whenever ${\mathfrak{a}}$ and ${I}$ satisfy

• ${\mathfrak{a}}$ is a progressive set of regular cardinals with ${\sup(\mathfrak{a})=\mu}$ and ${|\mathfrak{a}|<\theta}$,
• ${I}$ is a ${\sigma}$-complete ideal on ${\mathfrak{a}}$ extending the bounded ideal, and
• the structure ${(\prod\mathfrak{a}, <_I)}$ has a true cofinality.

To save ourselves some time later, let us agree to call such ${\mathfrak{a}}$ and ${I}$ relevant.

Now let ${\chi}$ be a sufficiently large regular cardinal, and let ${M}$ be an elementary submodel of ${\langle H(\chi),\in, <_\chi\rangle}$ satisfying

• ${|M|<\lambda}$,
• ${\{\sigma,\theta,\mu, \lambda\}\in M}$, and
• ${M\cap\lambda}$ is an ordinal of cofinality greater than ${\theta}$.

Our goal is to show that the collection

$\displaystyle \mathcal{P}:= M\cap [\mu]^{<\mu} \ \ \ \ \ (3)$

witnesses ${{\rm cov}(\mu,\mu,\theta,\sigma)\leq |M|}$, i.e., that every set in ${[\mu]^{<\theta}}$ can be covered by a union of fewer than ${\sigma}$ sets from ${\mathcal{P}}$.