${\mathrm{\Sigma }}_3^1$ sets in the Sacks model

We show that in the iterated Sacks model over the constructible universe the Mansfield-Solovay Theorem holds for ${\mathrm{\Sigma }}_3^1$ sets. In particular, every ${\mathbf{\Sigma }}_3^1$ set is Marczewski measurable and the optimal complexity for a Bernstein set is ${\mathrm{\Delta }}_4^1$. Based on a result by Kanovei, we also briefly show how to separate the Mansfield-Solovay Theorem at non-trivial levels of the projective hierarchy.