# Separation principle in generic models?

Separation principle $$\mathbf{Sep}(\mathbf\Sigma^1_n,\mathbf\Delta^1_n)$$ claims that any two disjoint pointsets of boldface class $$\mathbf\Sigma^1_n$$ are separated by a $$\mathbf\Delta^1_n$$ set. Separation principle $$\mathbf{Sep}(\mathbf\Pi^1_n,\mathbf\Delta^1_n)$$ is understood similarly. It is known classically that $$\mathbf{Sep}(\mathbf\Sigma^1_1,\mathbf\Delta^1_1)$$ and $$\mathbf{Sep}(\mathbf\Pi^1_2,\mathbf\Delta^1_2)$$ hold, while $$\mathbf{Sep}(\mathbf\Pi^1_1,\mathbf\Delta^1_1)$$ and $$\mathbf{Sep}(\mathbf\Sigma^1_2,\mathbf\Delta^1_2)$$ fail. The axiom $$V=L$$ (anyway, the existence of a good $$\mathbf\Delta^1_2$$ wellordering of the reals) implies that $$\mathbf{Sep}(\mathbf\Pi^1_n,\mathbf\Delta^1_n)$$ holds, while $$\mathbf{Sep}(\mathbf\Sigma^1_n,\mathbf\Delta^1_n)$$ fails for all $$n\ge3$$. The axiom of projective determinacy $$\mathbf{PD}$$ implies some other behaviour. But very little seems to be known about $$\mathbf{Sep}$$ in various generic models, including such best known ones like adding $$\kappa$$-many Cohen reals, $$\kappa\ge\omega_1$$. Basically, afaik this is 1) Harrington's unpublished handwritings of 1974-75, and my own APAL, 2016, 167, 3, 262–283. I wonder can someone share some other results in this direction.

Kanovei, Vladimir; Lyubetsky, Vassily, Counterexamples to countable-section $$\varPi_2^1$$ uniformization and $$\varPi_3^1$$ separation, Ann. Pure Appl. Logic 167, No. 3, 262-283 (2016) ZBL06529281 MR3437647