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

TL;DR

This paper constructs models of set theory demonstrating the existence of certain complex graphs on the real line that cannot be generated by countable families of projective or ROD functions, addressing questions in descriptive set theory.

Contribution

It introduces models where specific locally countable $oldsymbol{ m ext{Pi}}^1_2$ graphs are not generated by countably many projective or ROD functions, answering a question by Rettich and Serafin.

Findings

Existence of a locally countable $oldsymbol{ m ext{Pi}}^1_2$ graph not generated by countably many projective functions.

The $oldsymbol{ m ext{Sigma}}^1_2$ equi-constructibility graph on reals is not generated by countably many ROD functions in the Solovay model.

Provides models of set theory with these properties.

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.