Short··0:22·Combinatorics
Euclidean Ramsey Theory: OpenAI Claims a Full Classification
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About this video
A finite shape is Euclidean Ramsey if, however you colour space with finitely many colours, a same-size copy of it in one colour always appears once the dimension is high enough; shapes with three points in a line fail, so every Ramsey shape lies on a sphere. OpenAI claims a complete classification of Ramsey shapes by one algebraic (tensor) condition over their coordinates: every set of at most five points on a circle is Ramsey, and some cyclic quadrilaterals are Ramsey without being part of any symmetric (transitive) set, disproving the Leader–Russell–Walters conjecture. This is OpenAI's preprint, not yet peer reviewed.
Paper (OpenAI, catalogue result 172):
- A classification of finite Euclidean Ramsey configurations: github.com/openai/math/blob/main/preprints/A-classification-o…
The animation illustrates the result; it is not a proof.
Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/172
About the problem
Combinatorics·OpenAI Math·Result 172
Classification of finite Euclidean Ramsey configurations
Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set. (Lean)
Topics
Euclidean Ramsey theoryEuclidean Ramsey configurationsRamsey setsEuclidean Ramsey setsLeader Russell Walters conjecturesubtransitive setstransitive setscyclic quadrilateralRamsey theorycolouring spaceErdos Grahammonochromaticcombinatorial geometrydiscrete geometrycombinatoricsOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animation
