TITLE: Improving Baumgartner's result?
QUESTION [8 upvotes]: Q1: Is it consistent with the failure of CH to have an $\aleph_1$-dense subset $A \subseteq \mathbb{R}$ such that for every $X \subseteq \mathbb{R}$ of size $\aleph_1$, there is a $C^{\infty}$ map $F: \mathbb{R} \to \mathbb{R}$ sending $X$ into $A$?
Q2: Is it consistent with the failure of CH to have an $\aleph_1$-dense subset $A \subseteq \mathbb{C}$ such that for every $X \subseteq \mathbb{C}$ of size $\aleph_1$, there is an analytic function $F: \mathbb{C} \to \mathbb{C}$ sending $X$ into $A$?
What I know: If we replace $C^{\infty}$ by continuous, a positive answer to Q1 follows from PFA. If we replace analytic by continuous in Q2, a positive answer follows from MA.
Thanks!

REPLY [5 votes]: The answer to your first question is no. Proposition 9.4 in Abraham, Rubin and Shelah "On the consistency of some partition theorems..." implies that there are $\aleph_1$-dense sets $A, B \subseteq \mathbb{R}$ such that there is no $C^1$ function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f[A] \subseteq B$. See also Proposition 1.2 in M. Burke, "Entire functions mapping uncountable dense sets of reals onto each other monotonically". On the other hand, if you only consider $A$ and $B$ which are everywhere of second category, you get a positive answer. This is Theorem 2.1 in the second paper.