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Next define the {\\it Kneser Graph} $K(n,r)$ to be the graph with vertex set $\\binom{[n]}{r}$, the collection of $r$-subsets of an $n$ element set, and edge set $E = \\{ vw: v,w\\in \\binom{[n]}{r}, v\\cap w = \\emptyset \\}$. For fixed $r\\geq 4$ and $n\\rightarrow \\infty$ we show that $$B(K(n,r)"},"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":"1512.06937","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-22T02:58:24Z","cross_cats_sorted":[],"title_canon_sha256":"6860b4e1e63f16f03a0a987ca5ab1101cd8984a70f86142054a0e9b2722cb099","abstract_canon_sha256":"62781f8692747992965aa265ad423b6170f526cd08a309fb1dca791d23d5ef82"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:53.419441Z","signature_b64":"5MWwuLd0LUGd/FYrxvEXbhCRv7OWlEQrBIrJO7ZnQqawrUz4FNS4IoFW7bIwpcutZrpnSfZHt+GQ4gqy9F+SAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66641ebf01bbe9ebe5dd21634cb848194c02fd4c1294dbac218990aa185c0587","last_reissued_at":"2026-05-18T01:23:53.418903Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:53.418903Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Bandwidth of the Kneser Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Derrek Yager, Tao Jiang, Zevi Miller","submitted_at":"2015-12-22T02:58:24Z","abstract_excerpt":"Let $G = (V,E)$ be a graph on $n$ vertices and $f: V\\rightarrow [1,n]$ a one to one map of $V$ onto the integers $1$ through $n$. Let $dilation(f) =$ max$\\{ |f(v) - f(w)|: vw\\in E \\}$. Define the {\\it bandwidth} $B(G)$ of $G$ to be the minimum possible value of $dilation(f)$ over all such one to one maps $f$. Next define the {\\it Kneser Graph} $K(n,r)$ to be the graph with vertex set $\\binom{[n]}{r}$, the collection of $r$-subsets of an $n$ element set, and edge set $E = \\{ vw: v,w\\in \\binom{[n]}{r}, v\\cap w = \\emptyset \\}$. For fixed $r\\geq 4$ and $n\\rightarrow \\infty$ we show that $$B(K(n,r)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06937","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1512.06937","created_at":"2026-05-18T01:23:53.418990+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.06937v1","created_at":"2026-05-18T01:23:53.418990+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.06937","created_at":"2026-05-18T01:23:53.418990+00:00"},{"alias_kind":"pith_short_12","alias_value":"MZSB5PYBXPU6","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_16","alias_value":"MZSB5PYBXPU6XZO5","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_8","alias_value":"MZSB5PYB","created_at":"2026-05-18T12:29:32.376354+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MZSB5PYBXPU6XZO5EFRUZOCIDF","json":"https://pith.science/pith/MZSB5PYBXPU6XZO5EFRUZOCIDF.json","graph_json":"https://pith.science/api/pith-number/MZSB5PYBXPU6XZO5EFRUZOCIDF/graph.json","events_json":"https://pith.science/api/pith-number/MZSB5PYBXPU6XZO5EFRUZOCIDF/events.json","paper":"https://pith.science/paper/MZSB5PYB"},"agent_actions":{"view_html":"https://pith.science/pith/MZSB5PYBXPU6XZO5EFRUZOCIDF","download_json":"https://pith.science/pith/MZSB5PYBXPU6XZO5EFRUZOCIDF.json","view_paper":"https://pith.science/paper/MZSB5PYB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.06937&json=true","fetch_graph":"https://pith.science/api/pith-number/MZSB5PYBXPU6XZO5EFRUZOCIDF/graph.json","fetch_events":"https://pith.science/api/pith-number/MZSB5PYBXPU6XZO5EFRUZOCIDF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MZSB5PYBXPU6XZO5EFRUZOCIDF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MZSB5PYBXPU6XZO5EFRUZOCIDF/action/storage_attestation","attest_author":"https://pith.science/pith/MZSB5PYBXPU6XZO5EFRUZOCIDF/action/author_attestation","sign_citation":"https://pith.science/pith/MZSB5PYBXPU6XZO5EFRUZOCIDF/action/citation_signature","submit_replication":"https://pith.science/pith/MZSB5PYBXPU6XZO5EFRUZOCIDF/action/replication_record"}},"created_at":"2026-05-18T01:23:53.418990+00:00","updated_at":"2026-05-18T01:23:53.418990+00:00"}