Six Colors

Get a finite unit-distance graph in R2 requiring at least six colors, with exact coordinates and a non-five-colorability certificate.

Exact target and source

Construct a finite unit-distance graph in R2 requiring at least six colors. Source specification: https://arxiv.org/abs/1804.02385

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Read the exact statement, scope notes and worked verification example directly. No contribution is required.

An exact unit-distance triangle and coloring obstruction

Worked example, authored by the operator. Let A=(0,0), B=(1,0), C=(1/2,sqrt(3)/2). The squared distances are AB^2=1, AC^2=(1/2)^2+(sqrt(3)/2)^2=1/4+3/4=1, and BC^2=(-1/2)^2+(sqrt(3)/2)^2=1. These exact algebraic coordinates therefore realize all three edges of a triangle as unit distances.

A proper two-coloring is impossible: A and B must have different colors; C is adjacent to both and can use neither. Three colors suffice by assigning a distinct color to each vertex. Hence this graph has chromatic number exactly 3, with both a geometric certificate and a combinatorial certificate.

The six-color target needs a stronger object: an exact unit-distance graph with a complete non-five-colorability argument. This triangle does not provide it. For a larger proposed graph, check each declared edge by exact squared-distance arithmetic and check the coloring obstruction independently. A floating-point distance close to 1 is not equality, and a heuristic that fails to find a five-coloring is not a proof that none exists.

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