Why five colors would not suffice for the Euclidean plane

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Family 158 · Combinatorics
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

Color every point of the plane so that points exactly one unit apart always have different colors. How many colors are necessary? The picture may suggest colored regions, but the original problem allows much stranger assignments: a color class need not be measurable, connected or describable by a boundary.

This is the Hadwiger–Nelson problem. A classical construction uses seven colors, while a 2018 breakthrough showed that four cannot suffice. The manuscript in family 158 claims that five cannot suffice either, leaving only six and seven as possible answers.

Its central issue is the unrestricted nature of the coloring. Arguments about areas, density points or interfaces cannot simply be applied to arbitrary sets. The source proposes a bridge to a weaker measurable formulation and then derives a geometric contradiction for five labels.

2. Background and essential definitions

A proper $k$-coloring is a function

$$c:\mathbb R^2\longrightarrow\{1,\ldots,k\}$$

such that $c(x)\ne c(y)$ whenever $\|x-y\|=1$. The least possible $k$ is denoted $\chi(\mathbb R^2)$. Equivalently, form a graph with all plane points as vertices and unit-distance pairs as edges, and ask for its chromatic number.

This graph is not a planar graph in the graph-theoretic sense. Its straight edges may cross, so the Four-Color Theorem does not apply. A finite set of points with prescribed unit-distance edges can nevertheless obstruct a coloring of the entire plane: any whole-plane coloring restricts to it.

The distance one is a normalization. Rescaling the plane gives the same problem for any one fixed positive forbidden distance. Forbidding a whole interval of distances is a different, stronger constraint.

Proper versus weak measurable colorings

A measurable coloring has color classes to which ordinary area measure applies. M1 introduces a weaker condition than properness. Let $A_i$ be color class $i$, let $S^1$ be the unit circle, and let $\sigma$ be its normalized arc-length measure. A weak measurable coloring satisfies, for every $R>0$,

$$\sum_{i=1}^k\int_{B(0,R)}\int_{S^1} \mathbf1_{A_i}(x)\mathbf1_{A_i}(x+u) \,d\sigma(u)\,dx=0.$$

Here $B(0,R)$ is the disk of radius $R$, and $\mathbf1_A$ is one on $A$ and zero elsewhere. The nonnegative integrand records monochromatic unit pairs. The condition says these pairs have measure zero when both starting position and direction are sampled in the indicated way.

Exceptional pairs are permitted. Changing labels on an area-zero set does not affect this condition. A weak measurable coloring is therefore not being asserted to be a proper measurable coloring point by point.

3. Development history and previous partial results

The problem traces to Nelson in 1950, with early bounds of four and seven. The seven-color upper bound comes from repeating colors on a suitably sized hexagonal tiling. Same-color hexagons are separated far enough apart, while each individual hexagon has diameter below one.

Moser and Moser’s seven-vertex spindle supplied a particularly small obstruction to three colors in 1961. Its overlapping triangle constraints show that even a modest finite configuration can force a fourth color. The original publication is their solution to Problem 10.

In 2018, de Grey constructed a finite unit-distance graph that is not four-colorable, raising the unrestricted lower bound to five. The corrected construction has 1,581 vertices. Subsequent work reduced the size: Heule used satisfiability methods to produce 553-vertex examples, and Parts obtained a 509-vertex example. These developments concern finite obstructions to four colors, not the exclusion of five. See de Grey’s paper and Heule’s paper.

A separate line of work studied measurable or geometrically regular colorings. Falconer proved a measurable lower bound of five using density points. Work on planar maps and polygonal color regions gave stronger restrictions under additional geometric assumptions. Those assumptions matter: an argument about a tidy boundary does not automatically say anything about an arbitrary coloring.

The present manuscript aims to remove that gap in a specific way. It does not assume that an unrestricted coloring can itself be made measurable while preserving every unit-distance constraint. Instead, it claims equivalence of existence with the weak measurable condition defined above.

4. What the new manuscripts claim

M1’s Theorem 1.1 asserts, in the usual ZFC set-theoretic framework,

$$6\le\chi(\mathbb R^2)\le7.$$

There is no restriction on the color classes in this statement. The upper bound is the classical hexagonal construction; the new headline is the lower bound six.

Two further statements organize the argument. Theorem 1.3 says that, for every positive integer $k$, a proper unrestricted $k$-coloring exists if and only if a weak measurable $k$-coloring exists. Theorem 1.4 says there is no weak measurable five-coloring. Combining them gives the unrestricted conclusion.

The transfer theorem is an existence equivalence between two kinds of object. It is not a claim that every arbitrary coloring agrees almost everywhere with a measurable proper coloring, nor that the measurable chromatic number equals the unrestricted one.

The manuscript does not settle whether six colors suffice. It also does not display a small six-chromatic unit-distance graph. Graph compactness implies that an unrestricted obstruction to five colors has some finite witness, but finding an explicit manageable witness is a separate task.

5. Why the advance matters

Importance: 4/5. Raising the unrestricted plane-coloring lower bound would be a major advance on an exceptionally simple and persistent geometric question. It would reduce the possible answer to two adjacent integers.

The proposed transfer principle adds methodological significance. Measure theory provides powerful ways to discuss density and averaging, but those tools have often depended on regularity assumptions absent from the original problem. An exact existence bridge to the weak measurable condition would make them relevant without changing the question being answered.

The rating remains below five because the exact chromatic number is not determined by this claim. The remaining six-versus-seven distinction is still a substantial part of the original problem.

6. The main idea

Recover usable measurable information

The source first restricts a hypothetical coloring to the countable plane of complex algebraic numbers. Translating and rotating this coloring, then averaging, produces an invariant probability law on labelings. This introduces symmetry without requiring the original color classes to have an area.

A spectral analysis then separates components associated with ordinary continuous plane frequencies from a remaining part. In elementary Fourier analysis, frequencies are functions such as $x\mapsto e^{i\xi\cdot x}$; the source works in a larger character space and must distinguish these ordinary frequencies from its other characters.

Its rigidity theorem controls the remaining component. Projecting onto the continuous part preserves nonnegative label weights summing to one and retains the relevant zero correlations at unit distance. These weights can be represented by measurable fields. Choosing a positive label gives the weak measurable coloring.

The converse uses density points, locations where one measurable color fills an asymptotically full proportion of small surrounding disks. The manuscript shows how the weak exclusion controls unit pairs among these points. A common translation places any fixed finite configuration in suitable typical positions; compactness then recovers an unrestricted proper coloring of the whole graph.

Derive interfaces rather than assume them

For five labels, M1 studies the colors visible near circles of radius one. At a given direction, a limiting palette records the labels that can occur with positive weight. The source derives restrictions on these palettes and on transitions between them.

It next smooths color indicators by averaging over small disks. Away from controlled exceptional regions, the resulting label vectors use at most two colors at once. This allows a description in terms of edges of the complete graph on the five labels.

A topological argument forces a nontrivial loop of label transitions. Limits of the corresponding interfaces give connected sets meeting at a common point. Each connected set has two associated labels, and geometry excludes those labels from regions whose distances to the set lie on both sides of one.

Exhaust the possible label cycles

A simple cycle on five labels has length three, four or five. The manuscript treats each possibility separately. Five-cycles yield incompatible angular patterns; four-cycles produce a parity obstruction; three-cycles leave a region effectively restricted to three labels, where a carefully placed Moser spindle gives the contradiction.

The interfaces used in this reasoning are outputs of averaging and topology. The argument is not based on assuming polygonal color boundaries at the start. That is the key connection between its measure-theoretic first half and geometric second half.

7. Scope, limitations, and open questions

The conclusion concerns one exact forbidden distance and all plane points. Statements about polygonal colorings, measurable proper colorings or intervals of forbidden distances should not be substituted for it, even when they involve similar drawings.

The weak condition permits exceptional unit pairs, and its measure is the product of planar position and circular direction. It is not a statement that every circle has no monochromatic pairs or that every direction separately satisfies a pointwise exclusion.

The exact answer remains six or seven within the claimed result. Useful further questions include an explicit finite obstruction to five colors, bounds on its size, and whether the transfer method has analogues in other dimensions or distance graphs. These are not supplied by simply invoking compactness.

8. Sources and further reading

M1’s Theorems 1.1, 1.3 and 1.4 state the headline, transfer principle and weak measurable obstruction. Sections 2–4 develop the spectral and measurable passage; Sections 5–8 derive and exclude the possible label cycles. The distinction between weak and proper colorings is part of the theorem’s formulation.

The papers linked in Section 3 give the spindle, the 2018 finite-graph breakthrough and the measurable antecedent. Wikipedia’s Hadwiger–Nelson problem offers optional background illustrations; the exact conventions of this digest are stated above.

Source manuscripts

• M1. OpenAI, The Euclidean plane is not five-colorable (September 23, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.