Research paper
The arc chromatic number for Galois projective planes, affine planes and Euclidean grids
This paper by Gabriela Araujo-Pardo and Leonardo Martínez-Sandoval studies arc chromatic numbers in Galois projective planes, affine planes, and Euclidean grids.
After the first version of the work, I contacted the authors with a five-colouring of the 9×9 Euclidean grid G9 containing no monochromatic collinear triple. The colouring was independently checked by the authors. Together with the known lower bound, it establishes χ_A(G9) = 5.
The current public arXiv v2 records this as personal communication in its discussion of the open Euclidean-grid problem: the authors write that Haoxuan (Jason) Dong sent them a five-colouring of G9, which together with the known lower bound would settle χ_A(G9) = 5, and note that the colouring was independently verified.
The paper
The paper studies partitions into arcs in finite projective and affine planes and applies these ideas to Euclidean grids. It establishes exact results in several finite-geometric settings, gives asymptotic bounds for Euclidean grids, and determines exact values for a number of small cases.
The 9×9 five-colouring
While investigating the Euclidean-grid problem left open in the original paper, I found a five-colouring of G9 with no monochromatic collinear triple. I prepared the complete colouring together with independent verification programs and sent the construction to the authors for external checking.
Combined with the previously established lower bound, the colouring determines the exact value χ_A(G9)=5.
Independent verification and attribution
The authors independently checked the colouring after I contacted them. The current public arXiv v2 records this as personal communication in the discussion of the open problem: it states that Haoxuan (Jason) Dong sent the authors a five-colouring of G9 and notes that, together with the lower bound, it would settle χ_A(G9) = 5.
The paper also notes that I subsequently joined Thomas Prellberg, Ivet Lobo, and Matthew Lewis to continue investigating the problem.