Existence of a unit-distance graph with independence ratio strictly below 1/4 is established via a two-vertex augmentation of a prior 27-vertex construction, disproving a conjecture on geometric fractional chromatic number.
The chromatic number of the plane is at least 5
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abstract
We present a family of finite unit-distance graphs in the plane that are not 4-colourable, thereby improving the lower bound of the Hadwiger-Nelson problem. The smallest such graph that we have so far discovered has 1581 vertices.
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2026 1verdicts
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A unit-distance graph in the plane with independence ratio below 1/4
Existence of a unit-distance graph with independence ratio strictly below 1/4 is established via a two-vertex augmentation of a prior 27-vertex construction, disproving a conjecture on geometric fractional chromatic number.