Short··0:21·Combinatorics
Erdős Distinct Distances, Higher Dimensions: OpenAI Claims a Proof
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About this video
How few different distances can n points in space make? A grid of n points in d dimensions makes only about n^(2/d), and Erdős conjectured that no set can do much better. OpenAI claims a proof that for every fixed d ≥ 3, any n points in d-dimensional space determine at least c·n^(2/d) distinct distances, matching the grid up to a constant; this is OpenAI's preprint, not yet peer reviewed.
Paper (OpenAI, catalogue result 166):
- The higher-dimensional Erdős distinct-distances conjecture: github.com/openai/math/blob/main/preprints/The-higher-dimensi…
The animation illustrates the result; it is not a proof.
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Every result explained: gradientdamage.com/p/166
About the problem
Combinatorics·OpenAI Math·Result 166
The higher-dimensional Erdős distinct-distances conjecture
For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least cdn2/d distinct distances, with cd > 0 depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.
Topics
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