Color the Plane: The Euclidean plane cannot be colored with five colors

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Color the Plane

based on Result #158: The Euclidean plane cannot be colored with five colors

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	counting, coloring	Levels:	1
Category:	Combinatorics	Lean version:	YES! ✔
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The Euclidean plane cannot be colored with five colors. 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.

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LEVEL 1 The Euclidean plane is not five-colorable

released 2026-09-23 | 5 theorems · 32 lemmas · 44 proofs · 29,291 words | PLAY LEVEL 1 » (pdf)

We 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.

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