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Schrijver proved in 1978 that whenever $n\\ge 2k$, the chromatic number of the Kneser graph $\\mathrm{KG}\\big( \\binom{[n]}{k}_{2\\text{-stable}}\\big)$ is $n - 2k +2$. Generalizing this result, Meunier conjectured in 2011 that $\\chi\\left( \\mathrm{KG}\\big( \\binom{[n]}{k}_{s\\text{-stable}} \\big) \\right)= n - sk +s$ for all $n\\ge sk$. This conjecture was previo"},"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":"2607.12912","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-14T15:48:41Z","cross_cats_sorted":[],"title_canon_sha256":"6cb5daec473aa11656bac7bb72bbde62cf98844f3e38f2d1a4e4c94026660b2b","abstract_canon_sha256":"8eafd82afdc63a5e357191035ec97afd3c7294b7f44067e75e051ba38d5e94bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-15T01:22:24.806206Z","signature_b64":"5RhfYUkUk0lFQ5ITxrk/aRdewyOV2CbVtOitU2KO4+yS3hQR1RRN+abIOCYkSK1iqeAlUD5x/3oRHp3avM7ZAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f6c17f2982fd4897b4aaebfddc8adb82dc379d9a0a178afeffaccd189923b77","last_reissued_at":"2026-07-15T01:22:24.805231Z","signature_status":"signed_v1","first_computed_at":"2026-07-15T01:22:24.805231Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The chromatic number of 3-stable Kneser graphs","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Parker, Shira Zerbib, Wei-Chia Chen","submitted_at":"2026-07-14T15:48:41Z","abstract_excerpt":"For an integer $s \\ge 2$, a subset $S \\subseteq [n]$ is {\\em $s$-stable} if $\\min \\{j - i, n + i - j\\}\\ge s$ for every $i,j \\in S$ with $i<j$. Denote the set of all $s$-stable subsets of size $k$ of $[n]$ by $\\binom{[n]}{k}_{s\\text{-stable}}$. Schrijver proved in 1978 that whenever $n\\ge 2k$, the chromatic number of the Kneser graph $\\mathrm{KG}\\big( \\binom{[n]}{k}_{2\\text{-stable}}\\big)$ is $n - 2k +2$. Generalizing this result, Meunier conjectured in 2011 that $\\chi\\left( \\mathrm{KG}\\big( \\binom{[n]}{k}_{s\\text{-stable}} \\big) \\right)= n - sk +s$ for all $n\\ge sk$. 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