Paradoxical partitions of the reals by Robert Solovay

It holds in some generic extensions of the constructible universe $L$ (those by a single Sacks, Silver, Miller, Laver real) that there exists an OD (ordinal-definable) partition of the reals into two non-OD parts. The result for the Sacks extensions was announced by Solovay in 2002 but never published. The proofs combine forcing and descriptive set theoretic methods.