almost disjoint functions from $\aleph_{\omega+1}$ to $\aleph_\omega$

Question

Is it consistent that any collection of almost-disjoint functions $\aleph_{\omega+1}$ to $\aleph_\omega$ has size at at most $\aleph_{\omega+1}$?

"Almost-disjoint functions" are also called "eventually different functions." $f$ and $g$ are almost-disjoint with domain $\kappa$ when $\{ \alpha : f(\alpha) = g(\alpha) \}$ is bounded below $\kappa$.

For background, there are always $\kappa^+$ many almost-disjoint functions from $\kappa^+$ to $\kappa$. The Kurepa hypothesis for $\kappa^+$ implies there are $\kappa^{++}$ many a.d. functions. It is consistent (from large cardinals) that there are only $\aleph_1$ almost-disjoint functions from $\aleph_1$ to $\aleph_0$.

Answer 1

I see an answer. Thanks to Asaf for making me think about such things. Assume there is some stationary $S \subseteq \aleph_{\omega+1}$ such that $NS_{\aleph_{\omega+1}} \restriction S$ is $\aleph_{\omega+2}$-saturated. This is consistent by Foreman-Komjath, relative to a huge cardinal. (So this may be a very bad upper bound for the present question.)

Suppose there is a family $F$ of $\aleph_{\omega+2}$ many almost-disjoint functions from $\aleph_{\omega+1}$ to $\aleph_{\omega}$. For each $f \in F$, there is some stationary $S_f \subseteq S$ on which $f$ takes constant value $\alpha_f < \aleph_\omega$. There is a $G \subseteq F$ of size $\aleph_{\omega+2}$ such that all $f \in G$ have $\alpha_f =$ some constant $\beta$. So for $f,g \in G$, $S_f$ and $S_g$ are almost-disjoint stationary subsets of $S$. This contradicts saturation.

It would be interesting to see this from weaker assumptions.

Answer 2

Suppose $\kappa$ is a measurable cardinal that satisfies $\begin{pmatrix} \kappa^{++} \\ \kappa^+ \end{pmatrix} \to \begin{pmatrix} 2 \\ \kappa^+ \end{pmatrix}^{1,1}_\kappa$ (this implies the non-existence of almost disjoint functions from $\kappa^+$ to $\kappa$ of size $\kappa^{++}$). Now perform Prikry forcing $\mathbb{P}$ to change the cofinality of $\kappa$ to $\omega$. We claim in the final model $\begin{pmatrix} \kappa^{++} \\ \kappa^+ \end{pmatrix} \to \begin{pmatrix} 2 \\ \kappa^+ \end{pmatrix}^{1,1}_\kappa$ is true, where $\kappa$ is a strong limit of countable cofinality. Let $\dot{f}: \kappa^{++}\times \kappa^{+}\to \kappa$ be the name for the given coloring. Suppose $\Vdash_\mathbb{P} \neg \exists \alpha, \beta\in \kappa^{++} \exists B\in [\kappa^+]^{\kappa^+} f(\alpha, \cdot)\restriction B = f(\beta,\cdot)\restriction B$. For each $(\alpha,\beta)\in \kappa^{++}\times \kappa^+$, find $p_{\alpha,\beta}=(s_{\alpha,\beta}, A_{\alpha,\beta})\in \mathbb{P}$ that decides $\dot{f}(\alpha,\beta)$ to be $i_{\alpha,\beta}\in \kappa$. Define the following coloring in the ground: $(\alpha,\beta)\mapsto (i_{\alpha,\beta}, s_{\alpha,\beta})$. By the hypothesis, there exists $\alpha_0,\alpha_1\in \kappa^{++}$ and $B\in [\kappa^+]^{\kappa^+}$ such that the map gets constant value $(i,s)$ on $\{\alpha_0,\alpha_1\}\times B$. For each $\gamma\in B$, let $A_\gamma = A_{\alpha_0,\gamma}\cap A_{\alpha_1,\gamma}$. Fix any $\gamma\in B$, consider the following coloring on $[A_{\gamma}]^{<\omega}$: $\bar{v}\to 0$ if for cofinally many $\nu\in B$, $\bar{v}\subset A_{\nu}$, $1$ otherwise. By normality we can shrink it down to $A'\in U$ such that for all $n\in \omega$, $[A']^n$ get the same color, which must be $0$ (otherwise, for each tuple there is a tail of $\kappa^+$ that this tuple misses. But there are only $\kappa$ many tuples). Now find $(s^*, A^*)\leq (s, A')$ and $\sigma\in \kappa^+$ such that $(s^*, A^*)\Vdash \dot{f}(\alpha_0,\cdot)\neq \dot{f}(\alpha_1,\cdot)$ past $\sigma$. Since $s^*-s\subset A'$ we can find $\mu>\sigma$ in $B$ such that $s^*-s\subset A_\mu$, then $(s^*,A^*)\leq (s,A_\mu)$. But $(s,A_{\mu})\Vdash \dot{f}(\alpha_0,\mu)=i=\dot{f}(\alpha_1,\mu)$, contradiction. To get down to $\aleph_\omega$, we can interleave Lévy collapses with Prikry forcing.

On another note (i.e. no need to measurable nonsense), if you do $Coll(\aleph_{\omega+1}, <\kappa)$ for some inaccessible $\kappa$, then you'll have the negation of Kurepa Hypothesis at $\kappa^+$, which is the same as no almost disjoint functions from $\aleph_{\omega+1}\to \aleph_\omega$ of size $\aleph_{\omega+2}$.