The following problem was asked by Joe Miller in the fall of 2010 at a bar in Madison.

A subtree $T \subseteq 4^{< \omega}$ is $2$-bushy if for some node $\sigma \in T$, every node above $\sigma$ has two immediate successors. Is the following true: For every continuous function $f: 4^{\omega} \to 2^{\omega}$, there exists a $2$-bushy tree $T \subseteq 4^{< \omega}$ such that $f \upharpoonright [T]$ is either one-one or constant? Here $[T]$ is the set of branches through $T$.

I am tagging it set-theory + recursion-theory.