Liouville's theorem of complex analysis states that a bounded entire function is constant. I am trying to understand if a sort of converse holds in the following sense: consider a closed set $S \subset \mathbb{C}$ ($S$ may be unbounded) such that $\mathbb{C} \setminus S$ is unbounded. Can one construct a non-constant entire function $f$ such that $f$ is bounded on $S$ ?

More specifically, I am interested in understanding what types of unbounded closed subsets $S$ of $\mathbb{C}$ (if any) satisfy the following:

the complement $\mathbb{C} \setminus S$ is unbounded, and

All non-constant entire functions are unbounded on $S$.

Comment: I asked this question on MSE, but haven't got any answers there. With explicit examples involving trigonometric and exponential functions, one can construct rather "large" subsets of $\mathbb{C}$ where entire functions can be bounded (but my stock of examples is rather limited).