So, I think I’ve finished my “busy period”, and I have a little time to write. What I want to do today is post about the topic of my second lecture at this years Czech Winter School. The very first post I made on this blog dealt with an argument of Shelah that can be used to prove Solovay’s theorem that the Singular Cardinals Hypothesis holds above a strongly compact cardinal. The argument used a consequence of ${{\rm pp}(\mu)>\mu^+}$ as a “black box”; I want to take a look at the interior of this black box today.

Definition 1 Let ${\mu}$ a cardinal. We say that ${{\rm ADS}_\mu}$ holds if there is a sequence ${\langle A_\alpha:\alpha<\mu^+\rangle}$ such that each ${A_\alpha}$ is unbounded in ${\mu}$, and for each ${\beta<\mu^+}$, the collection ${\{A_\alpha:\alpha<\beta\}}$ is essentially disjoint, in the sense that there is a function ${F_\beta:\beta\rightarrow\mu}$ such that the collection ${\{A_\alpha\setminus F_\beta(\alpha):\alpha<\beta\}}$ is disjoint.

Some things to note:

• ${{\rm ADS_\mu}}$ always holds if ${\mu}$ is regular (any “almost disjoint” family of size ${\mu^+}$ has the required property).
• If ${\mu}$ is singular and ${{\rm ADS}_\mu}$ holds, then we may assume each ${A_\alpha}$ is of order-type ${{\rm cf}(\mu)}$.
• ${{\rm ADS_\mu}}$ for ${\mu}$ singular implies that that there is no countably complete uniform ultrafilter on ${\mu^+}$ — this is the content of the very first post I made on this blog.
• ${{\rm ADS_\mu}}$ follows from the existence of a better scale (and hence it follows from each of weak square and ${{\rm pp}(\mu)>\mu^+}$). The details of this are in my Handbook of Set Theory article.

What I want to do here is derive ${{\rm ADS_\mu}}$ from the assumption that ${\mu}$ is a singular cardinal of countable cofinality satisfying ${{\rm cov}(\mu,\mu,\aleph_1, 2)>\mu^+}$. Remember this means that for any family ${\mathcal{F}\subseteq[\mu]^{<\mu}}$ of cardinality ${\mu^+}$, there is a countable ${A\subseteq\mu}$ not covered by any member of ${\mathcal{F}}$. I’ll just state the theorem, and spread the proof out over the next couple of days in some more blog posts.

Theorem 2 (Shelah) Suppose ${\mu}$ is singular and ${{\rm cf}\mu=\aleph_0}$. If ${{\rm cov}(\mu,\mu,\aleph_1, 2)>\mu^+}$, then ${{\rm ADS}_\mu}$ holds.