This 28-page preprint, dated 22 September 2026, is by Jason Dong, Matthew Lewis, Ivet Lobo, Thomas Prellberg, and Ian Wanless. It studies the minimum number of colours needed to colour a square grid with no monochromatic collinear triple.
The paper proves that the arc chromatic numbers of the 11×11 and 12×12 grids are both 7. It also gives constructions and proofs for the values 5 for the 9×9 grid and 6 for the 10×10 grid.
Further results
The paper enumerates optimal colourings up to square symmetry and colour permutation for grids through 9×9, including 743 equivalence classes for the 9×9 grid.
It also proves that the 8p×8p grid can be coloured with at most 8p−8 colours for infinitely many primes p.
Availability
The 28-page preprint is publicly available in the project's GitHub repository.