Geodesic
Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes
Izvestiya. Mathematics , Tome 82 (2018) no. 1, pp. 61-90

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We use a countable-support product of invariant Jensen's forcing notions to define a model of $\mathbf{ZFC}$ set theory in which the uniformization principle fails for some planar $\varPi_2^1$ set all of whose vertical cross-sections are countable sets and, more specifically, Vitali classes. We also define a submodel of that model, in which there exists a countable $\varPi_2^1$ sequence of Vitali classes $P_n$ whose union $\bigcup_nP_n$ is not a countable set. Of course, the axiom of choice fails in this submodel.
Keywords: uniformization, forcing
Mots-clés : Vitali classes. 
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     author = {V. G. Kanovei and V. A. Lyubetsky},
     title = {Non-uniformizable sets of second projective level with countable cross-sections in the form of {Vitali} classes},
     journal = {Izvestiya. Mathematics },
     pages = {61--90},
     publisher = {mathdoc},
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     number = {1},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2018_82_1_a3/}
}
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V. G. Kanovei; V. A. Lyubetsky. Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes. Izvestiya. Mathematics , Tome 82 (2018) no. 1, pp. 61-90. http://geodesic.mathdoc.fr/item/IM2_2018_82_1_a3/