Projective Chromatic Numbers

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 $G$.