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 Basically, afaik this is 1) Harrington's unpublished handwritings of 1974-75, and my own APAL, 2016, 167, 3, 262–283.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


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