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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.03987","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-02T17:59:14Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"0761552c1a050b33fb1274fe7d49adebbd6ffa9f9d3cfb407d1d4802fe643fc4","abstract_canon_sha256":"e06ad932015d89300f13fb6a63b94ca4996ba28e97ffb05dc0c7221b31f71ff8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T02:06:08.802593Z","signature_b64":"F4H9nYW9MzcahaJ7xAU2FLQH9l+A0vj+W0lBeWGg6UORncZUBOnfVhYvNLi84obawyQXfhnNRYF1uwOwoqXrBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec10452ca1014611ddba8209937df030a15fdbb08278742703b115a19dd1a41c","last_reissued_at":"2026-06-03T02:06:08.802176Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T02:06:08.802176Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $1<p<2$. 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