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Level B · Reproducible Geometry P-max-to-min-ratios

Max to min ratios

Let n,d ≥ 2. Let C(d,n) denote the largest quantity such that, given any n distinct points x_1,…,x_n in R^d, the maximum distance max_1 ≤ i < j ≤ n ‖x_i-x_j‖ between the points is at least C(d,n) times the minimum distance min_1 ≤ i < j ≤ n ‖x_i-x_j‖. Establish upper and lower bounds for C(d,n).

From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-max-to-min-ratios,
  title        = {Max to min ratios},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/max-to-min-ratios}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

The problem

Let . Let denote the largest quantity such that, given any distinct points in , the maximum distance between the points is at least times the minimum distance . Establish upper and lower bounds for . What are the configurations that attain the minimal ratio between the two distances?

AlphaEvolve found a construction better than the previous record (at publication).

What counts as progress

  • A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
  • A proof that a known construction is optimal, or a better bound on the other side.
  • Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.