When a relation with all Borel sections will be Borel somewhere?

Chan, William; Magidor, Menachem

doi:10.1090/proc/15687

When a relation with all Borel sections will be Borel somewhere?
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by William Chan and Menachem Magidor PDF

Proc. Amer. Math. Soc. 150 (2022), 833-847 Request permission

Abstract:

In $\mathsf {ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every binary relation $R \in L(\mathbb {R})$ on $\mathbb {R}$ with all sections ${\mathbf {\Delta }_{1}^{1}}$ (${\mathbf {\Sigma }_{1}^{1}}$ or ${\mathbf {\Pi }_{1}^{1}}$) and every $\sigma$-ideal $I$ on $\mathbb {R}$ so that the associated forcing $\mathbb {P}_I$ of $I^+$ ${\mathbf {\Delta }_{1}^{1}}$ subsets is proper, there exists some $I^+$ ${\mathbf {\Delta }_{1}^{1}}$ set $C$ so that $R \cap (C \times \mathbb {R})$ is ${\mathbf {\Delta }_{1}^{1}}$ (${\mathbf {\Sigma }_{1}^{1}}$ or ${\mathbf {\Pi }_{1}^{1}}$, respectively).

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