**Problem 1.** Is it true that for every cardinal $\kappa\le\mathfrak c$ there exists a partition $(B_\alpha)_{\alpha\in\kappa}$ of the real line into $\kappa$ pairwise disjoint non-empty Borel subsets?

**Remark.** The answer to this problem is affirmative if $\mathfrak c\le \aleph_2$.

If the answer to Problem 1 is negative, then what about

**Problem 2.** Let $\mathcal P$ be a partition of the real line into Borel subsets. Is $|\mathcal P|=\mathfrak c$ or $|\mathcal P|\le\aleph_1$ in ZFC?