Let $f$: $\mathbb{R}\longrightarrow\mathbb{R}$ be an arbitrary function.
Must there exist $E\subseteq\mathbb{R}$ of cardinality $\aleph_1$, such that $f$ restricted to $E$ is monotonic?
Assuming CH, the answer is NO, as has been shown previously in Monotonic "Subfunction"
A positive answer follows from SOCA (semi open coloring axiom) which says the following: Whenever $X$ is an uncountable separable metric space and $U \subseteq [X]^2$ is open, there is a $Y \in [X]^{\omega_1}$ such that $[Y]^2$ is either contained in or disjoint with $U$. Assuming SOCA, you can easily show that every one-one uncountable function $f \subseteq \mathbb{R}^2$ is strictly monotone on some uncountable set. A good reference for SOCA is Abraham, Rubin, Shelah: On the consistency of some partition theorems for continuous colorings, and the structure of $\aleph_1$-dense real order types, Annals of pure and applied logic 29 (1985), 123-206.