{"paper":{"title":"Kusner's conjecture: Exact values and linear bounds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Hong-Jun Ge, Yang Zhou, Zixiang Xu","submitted_at":"2026-06-02T17:59:14Z","abstract_excerpt":"In 1983, Kusner conjectured that the largest equilateral set in $\\mathbb{R}^{n}$ with metric $\\ell_{p}$ has cardinality $n+1$ when $1<p<\\infty$ and $2n$ when $p=1.$ This conjecture was proved only in the isolated cases $p=2$ and $p=4$, and was disproved when $1<p<2$. The best general upper bound $O_p(n^{\\frac{2p+2}{2p-1}})$ is due to the celebrated work of Alon and Pudl\\'ak~[GAFA, 2003]. Our main contributions include:\n  (1) We prove Kusner's conjecture for every dimension $n\\ge 1$ when $2\\le p\\le 4$. More generally, for every integer $k\\ge 0$ and every $p\\in[4k+2,4k+4]$, every equilateral set"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.03987","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.03987/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}