Locally countable graphs of second projective class not generated by countably many projective functions

Vladimir Kanovei; Vassily Lyubetsky

paper

Locally countable graphs of second projective class not generated by countably many projective functions

arXiv:2605.03126

Abstract

To answer a question by Rettich and Serafin, we define a model of set theory in which there exists a locally countable $Π_{2}^{1}$ graph on a subset of the real line, which is not generated by a countable family of projective (or even real-ordinal definable, ROD) functions. We also prove that the $Σ_{2}^{1}$ equi-constructibility graph on the reals is not generated by a countable family of ROD functions in the Solovay model.

Hopefully final