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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.21813 |
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Table of Contents:
- 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 $G$.