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

Abstract

To answer a question by Rettich and Serafin, we define a model of set theory in which there exists a locally countable $\varPi^1_2$ 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 $\varSigma^1_2$ equi-constructibility graph on the reals is not generated by a countable family of ROD functions in the Solovay model.