What does this research mean for the field?

The chromatic number of the plane is bounded between 5 and 7, with the lower bound of 5 achieved by explicit 553-vertex unit-distance graphs. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Investigate the Hadwiger–Nelson problem to determine the bounds on the chromatic number of the Euclidean plane.

September 3, 2026Open Access

Unit-Distance Graphs: 553-Vertex Proofs Tighten Plane Chromatic Bound to 5,7 — E8 Intelligence Research

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Authors

ACAndrew Stewart Caldin

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Implication

Mathematical analysis reveals a five-color lower bound using 553-vertex unit-distance graphs in the Euclidean plane, highlighting a narrowed chromatic range between five and seven.

Key Points

Investigate the Hadwiger–Nelson problem to determine the bounds on the chromatic number of the Euclidean plane.
Constructed unit-distance graphs using the 7-vertex Moser spindle and 10-vertex Golomb graph as initial seeds via graph substitution.
Applied clausal proof minimization to extract minimal 553-vertex subgraphs.
Evaluated graph independence ratios under the condition α(G)/|V| ≤ 1/χ, with α(G) ≤ 0.2|V| required for χ = 5.
Established an explicit lower bound of χ(ℝ²) ≥ 5 using the minimal 553-vertex unit-distance graphs.
Maintained the known upper bound of 7 derived from hexagonal tiling, narrowing the chromatic number of the plane to χ(ℝ²) ∈ [5,7].

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9935b1636c6408cfa7e2db https://doi.org/10.5281/zenodo.22230093

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Implication

Mathematical analysis reveals a five-color lower bound using 553-vertex unit-distance graphs in the Euclidean plane, highlighting a narrowed chromatic range between five and seven.

Unit-Distance Graphs: 553-Vertex Proofs Tighten Plane Chromatic Bound to 5,7 — E8 Intelligence Research

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Implication

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Cite This Study

Unit-Distance Graphs: 553-Vertex Proofs Tighten Plane Chromatic Bound to 5,7 — E8 Intelligence Research

Discussion

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Implication

Key Points

Cite This Study

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