Uniformization of almost disjoint families

Question

Suppose $\mathcal{F} \subseteq \mathcal{P} (\omega) $ is an almost disjoint family and $\aleph_0 < \vert \mathcal{F} \vert = \kappa < 2^{\aleph_0} $. Is it consistent that for some such cardinal $\kappa$, we can uniformize every two-valued function on the family; that is, if $\mathcal{F} = \langle A_i \mid i < \kappa \rangle $ and for each $i < \kappa$ the function $f_i : A_i \to 2$ is either constant 1 or constant 0, we can find a total function $f: \omega \to 2$ that agrees with each function almost everywhere?

Shelah's Proper and Improper Forcing contains a positive result under the assumption that $\mathcal{F}$ is a $\textit{tree}$. I would like to know if there is something more general, at least for some special $\kappa$.

Answer 1

No, this is not consistent: there is (provably in ZFC) an almost disjoint family of size $\aleph_1$ and a two-valued function on that family such that the function cannot be uniformized in the way you've described.

To get such a family and function, we'll use a Hausdorff gap, or (more accurately) just an $(\omega_1,\omega_1)$-gap. An $(\omega_1,\omega_1)$-gap is a double sequence $\langle A_\alpha :\, \alpha < \omega_1 \rangle$, $\langle B_\alpha :\, \alpha < \omega_1 \rangle$ of subsets of $\omega$ such that

(1) the $A_\alpha$'s are almost increasing, in the sense that if $\alpha < \beta$ then $A_\alpha \subseteq^* A_\beta$ (where $\subseteq^*$ means that $A_\alpha \setminus A_\beta$ is finite).

(2) the $B_\alpha$'s are almost decreasing, in the sense that if $\alpha < \beta$ then $B_\alpha \supseteq^* B_\beta$.

(3) $A_\alpha \subseteq^* B_\beta$ for all $\alpha,\beta < \omega_1$.

(4) there is no $C \subseteq \omega$ such that $A_\alpha \subseteq^* C \subseteq^* B_\beta$ for all $\alpha,\beta < \omega_1$.

(Actually, there are two different kinds of sequences that are called "Hausdorff gaps" or $(\omega_1,\omega_1)$-gaps in various places. Some places use subsets of $\omega$ with the $\subseteq^*$ relation, as above, and some use functions $\omega \rightarrow \omega$ with the $\leq^*$ relation instead. The Wikipedia article I linked to takes the latter approach, but the former is more natural for this problem. But let me point out that given a gap in $(\omega^\omega,\leq^*)$, we can get one in $(\mathcal P(\omega),\subseteq^*)$ by just identifying a function with the set of points in $\omega \times \omega$ underneath its graph.)

Given a gap like this, let $\mathcal A$ be the family of all sets of the form $A_{\alpha+1} \setminus A_\alpha$ or $B_\alpha \setminus B_{\alpha+1}$. Let $f$ denote the function $\mathcal A \rightarrow 2$ that maps every set of the form $A_{\alpha+1} \setminus A_\alpha$ to $0$ and every set of the form $B_\alpha \setminus B_{\alpha+1}$ to $1$. Basically, $(3)$ implies that this is an AD family, and $(4)$ implies that there is no function $\omega \rightarrow \omega$ that uniformizes $f$. (If we had such a function, the preimages of $0$ and $1$ would "split the gap" and that can't happen.)