Projective Chromatic Numbers

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 $\mathbfΔ^1_n$-definable well-order of the reals implies $χ_{\mathbf{Δ^1_n}}(G) = χ(G)$ for all locally countable $\mathbf{Δ^1_n}$-definable graphs $G$, and that the presence of a $\mathbf{Δ^1_2}$-definable well-order of the reals implies $χ_{\mathbf{Δ^1_2}}(G) = χ(G)$ for all locally countable Borel graphs $G$.