Projective Chromatic Numbers

TL;DR

This paper explores the extension of classical graph colorability concepts to the projective setting, analyzing how definability and well-orderings influence chromatic numbers in higher projective pointclasses.

Contribution

It establishes that certain definable well-orderings of the reals imply equalities between definable and classical chromatic numbers for specific classes of graphs.

Findings

Presence of a -definable well-order of reals implies -definable chromatic number equals classical for locally countable graphs.

Presence of a 2-definable well-order of reals implies 2-definable chromatic number equals classical for Borel graphs.

Results extend classical graph colorability notions to higher projective pointclasses.

Abstract

We extend classical notions of definable colourability of graphs to the general projective setting and investigate whether known results, mainly about the $G_0$ dichotomy and the $2n + 1$ conjecture, hold in the context of higher projective pointclasses. We establish that for $n \ge 2$, the presence of a $\mathbf{\Delta}^1_n$-definable well-order of the reals implies $\chi_{\mathbf{\Delta^1_n}}(G) = \chi(G)$ for all locally countable $\mathbf{\Delta^1_n}$-definable graphs $G$, and that the presence of a $\mathbf{\Delta^1_2}$-definable well-order of the reals implies $\chi_{\mathbf{\Delta^1_2}}(G) = \chi(G)$ for all locally countable Borel graphs $G$.