The Hadwiger-Nelson problem: the Euclidean plane is not five-colorable

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Hadwiger, Ungeloste Probleme Nr. 40, Elemente der Mathematik 16 (1961) evidence for this event

Every coloring of $\mathbb{R}^2$ with five colors, with arbitrary (possibly non-measurable) color classes, has two points at distance $1$ of the same color. With the classical seven-coloring this gives $6\le\chi(\mathbb{R}^2)\le 7$. The route is a transfer theorem showing that a proper $k$-coloring exists if and only if a 'weak measurable' $k$-coloring (monochromatic unit pairs of measure zero) exists, for every finite $k$, followed by a geometric obstruction to weak measurable five-colorings. It does not decide between six and seven, and it does not give a finite non-five-colorable unit-distance graph (one exists by de Bruijn-Erdos compactness, but none is exhibited). parent of this event

VibeMathed record: The Hadwiger-Nelson problem: the Euclidean plane is not five-colorable evidence for this event

Lean proof: OAI/Geometry/PlaneColoring/Five.lean (no_proper_five_coloring) evidence for this event

Comparator statement: EuclideanFiveColor.lean evidence for this event

The Hadwiger-Nelson problem: the Euclidean plane is not five-colorable evidence for this event

The Hadwiger-Nelson problem: the Euclidean plane is not five-colorable parent of this event