Open problem, Combinatorics and discrete mathematics, Geometry, posed 1950

Hadwiger–Nelson problem

Open

Determine the chromatic number $\chi(\mathbb{R}^2)$ of the Euclidean plane—that is, the minimum number of colors needed to color every point of $\mathbb{R}^2$ such that no two points at Euclidean distance $\|x - y\|_2 = 1$ receive the same color.

Research frontier as of 2026

As of 2026, the chromatic number of the plane satisfies $5 \le \chi(\mathbb{R}^2) \le 7$ . The lower bound $\chi(\mathbb{R}^2) \ge 5$ was established by Aubrey de Grey (2018) and independently by Exoo and Ismailescu (2020), with the smallest known $5$ -chromatic unit-distance graph having $509$ vertices (Parts, 2020). For measurable colorings, Falconer (1981) and Bachoc–Passuello–Thiery (2015) studied the bounds $\chi_m(\mathbb{R}^2) \ge 5$ and $\chi_m(\mathbb{R}^2) \ge 6$ , while whether a finite $6$ -chromatic unit-distance graph exists in $\mathbb{R}^2$ remains open.

Best known results

The chromatic number of the plane is at least $5$ : $5 \le \chi(\mathbb{R}^2) \le 7$ (Aubrey de Grey, 2018).
The smallest known $5$ -chromatic unit-distance graph in $\mathbb{R}^2$ has $509$ vertices (Jaan Parts, 2020, following Heule's $510$ -vertex graph).

Tools and where they stop

Tool	Achieved	Where it stops
Algebraic unit-distance lattices and SAT solvers	Embeds dense subgraphs of $\mathbb{Q}[\sqrt{3}, \sqrt{11}]$ with many Moser spindles and verifies via SAT solvers that every $4$ -coloring is blocked, proving $\chi(\mathbb{R}^2) \ge 5$ .	For $5$ -colorings, unit-distance graphs in $\mathbb{R}^2$ have maximum average degree too small to prevent $5$ -colorability without astronomically larger vertex sets.
Harmonic analysis and semidefinite programming for measurable colorings	Bounds the maximum density of measurable independent sets in $\mathbb{R}^2$ using Bessel functions, proving $\chi_m(\mathbb{R}^2) \ge 5$ .	Cannot rule out non-measurable $5$ -colorings or $6$ -colorings constructed via the axiom of choice.

Open questions

Is the chromatic number of the Euclidean plane $\chi(\mathbb{R}^2)$ equal to $5$ , $6$ , or $7$ ?
Does the value of $\chi(\mathbb{R}^2)$ depend on the axiom of choice even among models of ZFC?

References

Aubrey D. N. J. de Grey (2018). The chromatic number of the plane is at least 5 · arXiv:1804.02385
Alexander Soifer (2009). The Mathematical Coloring Book · DOI:10.1007/978-0-387-74642-5