**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?



Partitions of the real line into Borel subsets
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



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



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> W. Brian and A. W. Miller, "Partitions of $2^\omega$ and completely ultrametrizable spaces," *Topology and Its Applications* **184** (2015), pp. 61-71 ([link](https://wrbrian.files.wordpress.com/2012/01/ucr.pdf)).
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