The chromatic number of finite projective spaces

TL;DR

This paper investigates the chromatic number of finite projective spaces, establishing recursive bounds, refining results for q=2, and connecting to multicolor Ramsey numbers to improve lower bounds.

Contribution

It introduces a recursive upper bound for the chromatic number of finite projective spaces and links it to multicolor Ramsey numbers, providing new bounds and resolving an open case.

Findings

Established a recursive upper bound for hi_q(n)

Refined bounds for hi_2(n) and proved tightness for n  7

Connected hi_q(t;n) to multicolor Ramsey numbers and improved lower bounds

Abstract

The chromatic number of the finite projective space $\mathrm{PG}(n-1,q)$, denoted $\chi_q(n)$, is the minimum number of colors needed to color its points so that no line is monochromatic. We establish a recursive upper bound $\chi_q(n) \leq \chi_q(d) + \chi_q(n - d)$ for all $1 \leq d < n$ and use it to prove new upper bounds on $\chi_q(n)$ for all $q$. For $q = 2$, we further refine this recursion and prove that \[ \chi_2(n) \le \lfloor 2n/3 \rfloor + 1 \] for all $n \ge 2$, and that this bound is tight for all $n \le 7$. In particular, this recovers all previously known cases for $n \le 6$ and resolves the first open case $n = 7$. On the lower-bound side, using a connection with multicolor Ramsey numbers for triangles, we note that \[ \chi_2(n) \ge (1 - o(1))\,\frac{n}{\log n}. \] We also consider $\chi_q(t;n)$, the minimum number of colors needed to color the points of $\mathrm{PG}(n-1,q)$ with no monochromatic $(t - 1)$-dimensional subspace, and establish an equivalence between $\chi_q(t;n)$ and the multicolor vector-space Ramsey numbers $R_q(t;k)$. Using this equivalence together with our upper bounds on $\chi_q(t;n)$, we improve, for every fixed $t$, the best known lower bounds on $R_q(t;k)$ from $\Omega_q(\log k)$ to $\Omega(k)$.