A question of Erdős on entire functions At the end of the following paper, Erdős asked if there is a family $F$ of entire functions of size continuum such that for every $z \in \mathbb{C}$, $\{f(z) : f \in F\}$ has size less than continuum. He also showed how to construct such a family under CH.
Did someone solve it?
 A: Maybe following theorem help us to answer the question.
Theorem: The continuum hypothesis is true if and only if there is an
uncountable family $\mathcal{F}$ of entire analytic functions such that for each $z\in\mathbb{C}$ the set of values $\{f(z):z\in\mathbb{C}\}$ is countable.
Please see Theorem 14.4 of Forcing for Mathematicians  by Weaver for the proof
A: The following (negative answer to Erdos' question) will appear in a joint work with Shelah.
Claim: Suppose $V \models 2^{\aleph_0} = \lambda > \kappa = \aleph_1$. Let $P$ add $\kappa$ Cohen reals. Then in $V^{P}$, there is no such family.
Proof: Let $r \in {}^{\kappa}2$ be the Cohen generic sequence. Clearly $V[r] \models 2^{\aleph_0} = \lambda$. Suppose $\langle f_{\alpha} : \alpha < \lambda \rangle$ is a sequence of pairwise distinct analytic functions in $V[r]$. Choose $X \in [\lambda]^{\lambda}$, $\xi_{\star} < \kappa$ such that for each $\alpha \in X$, $f_{\alpha}$ is coded in $V[r \upharpoonright \xi_{\star}]$. Let $z_{\star} \in \mathbb{C}$ be Cohen over $V[r \upharpoonright \xi_{\star}]$. Since two distinct analytic functions only agree on a countable set, it follows that $\langle f_{\alpha}(z_{\star}) : \alpha \in X \rangle$ are pairwise distinct.
Update: Shelah and I showed that the answer to Erdos' question is independent of ZFC + not CH so the other direction also holds. 
An interesting question that remains open is: Is the following consistent: $2^{\aleph_0} = \aleph_2$ and there exists $U \in [\mathbb{C}]^{\aleph_1}$ such that for every $A \in [\mathbb{C}]^{\aleph_1}$, there is a non constant entire map that sends $A$ into $U$? 
