Euclidean Ramsey theory is the study of finite configurations of points whose congruent copies must occur monochromatically under finite colorings of sufficiently high-dimensional Euclidean space. A nonempty finite set is a Ramsey set if, for every positive integer , there is an such that every -coloring of contains a monochromatic congruent copy of at its original scale (Erdős et al. 1973). The colorings need not be measurable.

A spherical set lies on a sphere. A subtransitive set is congruent to a subset of a finite set on which its group of isometries acts transitively. Graham's spherical conjecture, recorded by Erdős et al. (1973), asks whether every finite spherical set is Ramsey.

Pálvölgyi (2026) reported a seven-point counterexample. For any transcendental number , the set

which lies on the circle , is claimed to be non-Ramsey. ChatGPT supplied the proofs and ideas. The proof uses a nonmeasurable derivation of and a finite-coloring theorem. As of Oct. 8, 2026, independent review of the claim had not been reported, and the author's disclosure explicitly excluded verification of further claims in the appendix.

Hoff et al. (2026) report that every red-blue coloring of the Euclidean plane with no red pair at distance 1 contains seven blue collinear points with consecutive spacing 1. Their proof uses a rotation-invariant-measure theorem from an OpenAI manuscript on the Hadwiger-Nelson problem, together with computational inequalities.

See also

Congruent, Euclidean Space, Ramsey Theory, Sphere

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References

Erdős, P.; Graham, R. L.; Montgomery, P.; Rothschild, B. L.; Spencer, J.; and Straus, E. G. "Euclidean Ramsey Theorems. I." J. Combin. Theory Ser. A 14, 341-363, 1973. https://doi.org/10.1016/0097-3165(73)90011-3.Hoff, R.; Hu, G.; and Liu, Y. "Euclidean Ramsey Theory for Lines and Finite Configurations." 10 Nov 2026. https://arxiv.org/abs/2611.14131v1.Pálvölgyi, D. "A Cyclic Non-Ramsey Heptagon." 20 Sep 2026. https://arxiv.org/abs/2609.23327.VibeMathed. "The Classification Problem for Finite Euclidean Ramsey Sets." Oct. 7, 2026. https://vibemathed.com/problem/euclidean-ramsey-sets-classification.

Cite this as:

Weisstein, Eric W. "Euclidean Ramsey Theory." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EuclideanRamseyTheory.html

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