Putting $\omega$ into $\alpha$ boxes where $\alpha \in \big(\omega\cup\{\omega\}\big)\setminus\{0,1\}$

Question

This is a follow-up question to an older question.

Let $\alpha \in \big(\omega\cup\{\omega\}\big) \setminus \{0,1\}$ be an ordinal. We say that a function $f: \omega \to \alpha$ is fair if $$|f^{-1}(\{j\})| = \aleph_0$$ for all $j\in\alpha$.

We call a set ${\frak F}$ of fair functions $f:\omega\to\alpha$ equalising for $\alpha$ if for all $a,b\in\omega$ there is $f\in{\frak F}$ with $f(a) = f(b)$. Let $B_\alpha$ be the minimum cardinality that an set equalising for $\alpha$ can have. This post establishes $B_2 \leq 3$.

Question. Can an explicit value for $B_\alpha$ be given for $\alpha \in \big(\omega\cup\{\omega\}\big) \setminus \{0,1\}$? I am particularly interested in $B_\omega$.

Answer 1

$B_\alpha = 3$ for every $\alpha \in (\omega \cup \{\omega\}) \setminus \{0,1\}$. Here is a construction that shows $B_\alpha \le 3$.

1. Let $g : \mathbb{N} \to \alpha$ be any function such that $g^{-1}(\{j\})$ is infinite for all $j \in \alpha$.
2. Let $h$ be any bijection from $\omega$ to $\{0,1,2\} \times \mathbb{N}$.
3. For $i \in \{0,1,2\}$ define $t_i : \{0,1,2\} \times \mathbb{N} \to \alpha$ as follows: $$t_i(r,s) = \begin{cases} g(s) & \text{if } i=r \\ 0 & \text{otherwise} \end{cases}$$ and define $f_i : \omega \to \alpha$ by $f_i = t_i \circ h$.

The functions $f_i$ are fair. In fact, $f_i^{-1}(\{j\}) = h^{-1} (t_i^{-1} (\{j\}))$, which is an infinite set because $h$ is a bijection and $t_i^{-1}(\{j\})$ contains $\{i\} \times g^{-1}(\{j\})$. The set $\{f_0,f_1,f_2\}$ is equalising for $\alpha$ because for all $a,b \in \omega$, if $h(a) = (r_a,s_a)$ and $h(b) = (r_b,s_b)$, then for any $i \not\in \{r_a,r_b\}$ we have $f_i(a)=f_i(b)=0$.

The reasoning used to show $B_2 > 2$ in the answer to this question extends to show $B_\alpha > 2$ for all $\alpha > 1$. Hence $B_\alpha = 3$ for all $\alpha \in (\omega \cup \{\omega\}) \setminus \{0,1\}$.