1 · The essentials
Imagine assigning a color to every point of a flat, endless plane. The rule is that any two points exactly one unit apart must have different colors. The main result says that five colors cannot satisfy this rule: somewhere, two points one unit apart will share a color. This holds however the colors are assigned, with no requirement that the colored sets have tidy shapes or patterns. The main result has a Lean formalization. Combined with the classical arrangement using seven colors, it leaves only two possibilities for the number of colors needed: six or seven.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine placing colored marks on an endless sheet of paper while checking pairs with a ruler of fixed length. With only five colors, the result says that a matching pair exactly that ruler-length apart is unavoidable. The image is limited because the mathematical task colors every point, not just the marks you can draw.
Why it matters
This narrows the possibilities in the Hadwiger–Nelson problem, which asks how many colors are needed to keep points one unit apart differently colored. The conclusion applies to every assignment of five colors, not merely to assignments with some prescribed geometric structure. It does not settle whether six colors suffice.
2 · Going further
The precise question concerns a coloring of the Euclidean plane: a function assigning each point one of a fixed number of colors. A valid coloring must assign different colors to any points and with . The main manuscript states that every assignment of five colors contains a monochromatic unit-distance pair, without any regularity assumption on the color classes. Thus, writing for the chromatic number of the plane, the result gives .
The repository summary places this result in the Hadwiger–Nelson problem and supplies the classical seven-coloring as the upper bound . Together these give . The main result has a Lean formalization. The supplied abstract states the conclusion but does not describe the argument, so no construction or proof strategy can be explained from this source. Only the main manuscript is supplied; no companion manuscripts or their relationships are described.
Useful notions
- Points and distances in the Euclidean plane
- Assigning colors to points
- A fixed unit of length
- Lower and upper bounds
What was known before / what is new
Before
The source identifies a classical seven-coloring of the plane, establishing that seven colors suffice. It places the question within the Hadwiger–Nelson problem but gives no further historical account or earlier lower bounds.
New
The main manuscript rules out every five-coloring of the Euclidean plane that would keep all unit-distance pairs differently colored. Its statement imposes no regularity assumption on the color classes. Together with the stated seven-color upper bound, this restricts the chromatic number to six or seven.
Limits and assumptions
The result concerns the entire Euclidean plane and pairs at distance exactly one, with no restriction on the color classes. It excludes five colors but does not determine whether six colors suffice. The main result has a Lean formalization; the supplied material does not describe its implementation or specify separate formal verification of the classical seven-coloring. The abstract provides no details of the argument.
Mini glossary
- Euclidean plane —
- The flat, two-dimensional plane with ordinary Euclidean distance.
- Color class —
- The set of all points assigned a particular color.
- Monochromatic unit-distance pair —
- Two points that have the same color and are exactly one unit apart.
- Chromatic number of the plane —
- The smallest number of colors needed to color every point so that points one unit apart always have different colors.
Keywords : Hadwiger–Nelson problem · Plane coloring · Unit distance
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
The Euclidean plane is not five-colorable
23/09/2026We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.