Ho 2026 erdos s diameter conjecture separated distances

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Boon Suan Ho, Erdős's diameter conjecture for separated distances fails in high dimensions. arXiv preprint (16 April 2026), 6 pages. arXiv:2604.15305v1, doi:10.48550/arXiv.2604.15305.

The held PDF is arXiv:2604.15305v1, and the statements below were checked against its page images. Erdos conjectured that an n-point set in Euclidean space in which any two of the n(n-1)/2 distances differ by at least 1 has diameter at least (1+o(1)) n^2, whatever the dimension (p. 1, display (1); the paper cites Problem 20 of Erdos's 1997 problem chapter and Problem 670 of erdosproblems.com). Ho disproves this (Theorem 1, p. 1) by constructing, for each prime power q, a set of n = q+1 points in dimension q^2+q with mutually 1-separated distances but diameter at most (1 - 1/pi^2 + o(1)) n^2; the proof is fully formalized in Lean 4. Proposition 8 (p. 5) evaluates the diameter of the construction as (1 - 1/pi^2 + o(1)) n^2. The acknowledgements (p. 6) credit GPT-5.4 Pro with finding the construction and Harmonic's Aristotle, assisted by GPT-5.4 Pro, with the Lean formalization. Remark 9 (p. 6) leaves the question open in each fixed dimension d >= 2, with (1+o_d(1)) n^2 in place of (1+o(1)) n^2. The paper was checked as a candidate recent result for problem 100 and the review confirmed it unrelated, since it refutes a different high-dimensional diameter conjecture rather than addressing the planar diameter question of that problem.

Source: https://arxiv.org/abs/2604.15305. The arXiv record (https://arxiv.org/abs/2604.15305, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #670: Theorem 1 answers the question in the negative when the dimension may grow with n (it uses dimension n^2 - n); in each fixed dimension the question stays open (Remark 9).