Erdős's diameter conjecture for separated distances fails in high dimensions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911599231500288 |
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| author | Ho, Boon Suan |
| author_facet | Ho, Boon Suan |
| contents | Erdős asked whether every $n$-point set in Euclidean space whose $\binom{n}{2}$ pairwise distances are mutually at least $1$ apart must have diameter at least $(1+o(1))n^2$. We disprove this statement by constructing for every prime power $q$ a set $\mathcal X_q\subset \mathbb R^{q^2+q}$ of $n=q+1$ points such that all pairwise distances in $\mathcal X_q$ are mutually at least $1$ apart, while $$\operatorname{diam}(\mathcal X_q)\le\Bigl(1-\frac{1}{π^2}+o(1)\Bigr)n^2.$$ The proof is fully formalized in Lean 4. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_15305 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Erdős's diameter conjecture for separated distances fails in high dimensions Ho, Boon Suan Combinatorics Metric Geometry 52C10 Erdős asked whether every $n$-point set in Euclidean space whose $\binom{n}{2}$ pairwise distances are mutually at least $1$ apart must have diameter at least $(1+o(1))n^2$. We disprove this statement by constructing for every prime power $q$ a set $\mathcal X_q\subset \mathbb R^{q^2+q}$ of $n=q+1$ points such that all pairwise distances in $\mathcal X_q$ are mutually at least $1$ apart, while $$\operatorname{diam}(\mathcal X_q)\le\Bigl(1-\frac{1}{π^2}+o(1)\Bigr)n^2.$$ The proof is fully formalized in Lean 4. |
| title | Erdős's diameter conjecture for separated distances fails in high dimensions |
| topic | Combinatorics Metric Geometry 52C10 |
| url | https://arxiv.org/abs/2604.15305 |