- How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
- How many subsets of the long line $\omega_1\times[0,1)$ are order isomorphic to $\mathbb{Q}$?

I can see that results in both cases are between $\mathfrak{c}$ and $\mathfrak{2^c}$.