The chromatic number of the finite projective space PG(n−1,q), denoted χ_q(n), is the minimum number of colors needed to color its points so that no line is monochromatic. We establish a new recursive bound, and using this recursion, we obtain new upper bounds on χ_q(n) for all q.
For q = 2, we refine the recursion and prove that χ_2(n) ≤ ⌊2n/3⌋ + 1 for all n ≥ 2, and that this bound is tight for all n ≤ 7. This recovers all previously known cases for n ≤ 6 and resolves the first open case n = 7.
We also make connections to the multicolor Ramsey numbers for triangles and multicolor vector-space Ramsey numbers.
This work is in collaboration with Anurag Bishnoi and Wouter Cames van Batenburg. Here is the arXiv link to the paper: https://arxiv.org/abs/2512.01760
