TITLE: Partitions of the real line into Borel subsets
QUESTION [17 upvotes]: 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?

REPLY [14 votes]: The answer to both problems is no!
If the Cohen forcing is used to add lots of reals to a countable transitive model of GCH, then in the resulting extension, any partition of the real line into Borel sets has size $\leq \aleph_1$ or $\mathfrak{c}$. (But the continuum can be anything with uncountable cofinality.) This is a theorem of Arnie Miller, Theorem 3.7 in

A. W. Miller, "Infinite combinatorics and definability," Annals of Pure and Applied Mathematical Logic 41 (1989), pp. 179-203.

On the other hand, it is consistent with any permissible value of $\mathfrak{c}$ that there is a partition of the real line into $\kappa$ closed sets, for every uncountable $\kappa \leq \mathfrak{c}$. This is a joint theorem of myself and Arnie Miller, Theorem 3.11 in

W. Brian and A. W. Miller, "Partitions of $2^\omega$ and completely ultrametrizable spaces," Topology and Its Applications 184 (2015), pp. 61-71 (link).