Continuing the construction, let us assume that ${\mu}$ is singular of countable cofinality and ${{\rm cov}(\mu,\mu,\aleph_1,2)>\mu^+}$. For each ${\beta<\mu^+}$, let ${\langle A^\beta_n:n<\omega\rangle}$ be a sequence of sets such that

• ${A^\beta_0=\emptyset}$,
• ${\beta=\bigcup_{n<\omega}A^\beta_n}$,
• ${|A^\beta_n|<\mu}$ for all ${n<\omega}$, and
• ${A^\beta_n\subseteq A^\beta_{n+1}}$.

By induction on ${\alpha<\mu^+}$ we choose sets ${x_\alpha\in[\mu]^{\aleph_0}}$ such that for no ${\beta<\mu^+}$ and ${n<\omega}$ is ${x_\alpha}$ a subset of ${\bigcup\{x_\gamma:\gamma\in A^\beta_n\cap\alpha\}}$.

Notice that each set of the form ${\bigcup\{x_\gamma:\gamma\in A^\beta_n\cap\alpha\}}$ is of cardinality less than ${\mu}$, and there are ${\mu^+}$ sets of this form. Our assumption that ${{\rm cov}(\mu,\mu,\aleph_1,2)}$ is greater than ${\mu^+}$ tells us that a suitable ${x_\alpha}$ can always be found.

Let ${\eta_\alpha:\omega\rightarrow x_\alpha}$ be a bijection, and let ${A_\alpha:=\{\eta_\alpha\upharpoonright\ell:\ell<\omega\}}$. We will show that the collection ${\{A_\alpha:\alpha<\mu^+\}}$ will witness ${{\rm ADS}_\mu}$.

What this amounts to is that for each ${\beta<\mu^+}$, we need a function ${h_\beta:\beta\rightarrow\omega}$ so that

$\displaystyle \Delta(\alpha,\gamma)\leq\max\{h_\beta(\alpha),h_\beta(\gamma)\} \ \ \ \ \ (1)$

for all ${\alpha,\gamma<\beta}$, where

$\displaystyle \Delta(\alpha,\gamma)=\text{ least }\ell\text{ such that }\eta_\alpha(\ell)\neq\eta_\gamma(\ell). \ \ \ \ \ (2)$

Lemma 1 For each ${n<\omega}$, the family ${\{x_\alpha:\alpha\in A^\beta_n\}}$ has a transversal (i.e., a one-to-one choice function) ${f^\beta_n}$.

Proof: We define ${f^\beta_n\upharpoonright (A^\beta_n\cap\alpha)}$ by induction on ${\alpha}$. This is trivial for ${\alpha=0}$ and ${\alpha}$ limit. If ${\alpha=\gamma+1}$, then ${x_\gamma}$ is not a subset of ${\bigcup\{x_\epsilon:\epsilon\in A^\beta_n\cap\gamma\}}$ so ${f^\beta_n(\gamma)}$ can be defined easily. $\Box$

Armed with the preceding lemma, we define a function ${k_\beta:\beta\rightarrow\omega}$ as follows:

Given ${\alpha<\beta}$, we know ${\alpha\in A^\beta_{n+1}\setminus A^\beta_n}$ for some unique ${n<\omega}$. We define ${k_\beta(\alpha)}$ to be the unique ${k<\omega}$ such that ${f^\beta_n(\alpha)\eta_\alpha(k)}$.

In English, ${k_\beta(\alpha)}$ answers the question “when does ${\eta_\alpha}$ enumerate the value of ${f^\beta_n(\alpha)}$?”.

Our goal is to show that the function ${k_\beta}$ almost works, and then show how to modify ${k_\beta}$ to a function ${h_\beta}$ that actually does the job.