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This implies $\\sup_{s\\ge1}w(s)^{1/s}=4$, as conjectured by Kohayakawa (1991). His recursive construction then gives induced paths of order $\\Omega(4^r/r^{5/2})$ in the Kneser graph $KG(2r+1,r)$ and yields \\[\n  \\max\\{\\cc(\\overline{P_n}),\\ \\cc(\\overline{C_n})\\}\n  \\le \\log_2 n+\\frac52\\log_2\\log_2 n+O(1). \\] Together w"},"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":"2608.11132","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-11T16:52:39Z","cross_cats_sorted":[],"title_canon_sha256":"a8709e6fff4382e7ee7a7c10e9001cc47adf78f4447ddcc494f6b59007aecd63","abstract_canon_sha256":"a97ca7504ef1a126f9cf70ddd17480bed2c29373793b709e8e730b5235cebf90"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-12T01:24:39.923841Z","signature_b64":"LGYtSl84IwiZRekRrN0EE5hHcaAYpr9uZYOnvt6pYY0HX5gPc1Qizq0jMVgGMkxLJ5gxB3naDnYGZEfTX2cbDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"984c4189993c90e9231149797d03307f8d6526b64f36dac8170fa829be4e23b1","last_reissued_at":"2026-08-12T01:24:39.922172Z","signature_status":"signed_v1","first_computed_at":"2026-08-12T01:24:39.922172Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Kohayakawa's conjecture and clique coverings of complements of paths and cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning","submitted_at":"2026-08-11T16:52:39Z","abstract_excerpt":"For $s\\ge1$, let $G_s$ be the bipartite graph between the $s$-subsets and the $(s-1)$-subsets of $[2s]$, where adjacency means disjointness, and let $w(s)$ be the maximum number of $s$-subsets on an induced path in $G_s$. We prove $w(s)\\ge \\frac{4^s}{2048s^{5/2}}$ for all $s\\geq 6$. This implies $\\sup_{s\\ge1}w(s)^{1/s}=4$, as conjectured by Kohayakawa (1991). His recursive construction then gives induced paths of order $\\Omega(4^r/r^{5/2})$ in the Kneser graph $KG(2r+1,r)$ and yields \\[\n  \\max\\{\\cc(\\overline{P_n}),\\ \\cc(\\overline{C_n})\\}\n  \\le \\log_2 n+\\frac52\\log_2\\log_2 n+O(1). \\] Together w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.11132","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/2608.11132/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2608.11132","created_at":"2026-08-12T01:24:39.926025+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.11132v1","created_at":"2026-08-12T01:24:39.926025+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.11132","created_at":"2026-08-12T01:24:39.926025+00:00"},{"alias_kind":"pith_short_12","alias_value":"TBGEDCMZHSIO","created_at":"2026-08-12T01:24:39.926025+00:00"},{"alias_kind":"pith_short_16","alias_value":"TBGEDCMZHSIOSIYR","created_at":"2026-08-12T01:24:39.926025+00:00"},{"alias_kind":"pith_short_8","alias_value":"TBGEDCMZ","created_at":"2026-08-12T01:24:39.926025+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/TBGEDCMZHSIOSIYRJF4X2AZQP6","json":"https://pith.science/pith/TBGEDCMZHSIOSIYRJF4X2AZQP6.json","graph_json":"https://pith.science/api/pith-number/TBGEDCMZHSIOSIYRJF4X2AZQP6/graph.json","events_json":"https://pith.science/api/pith-number/TBGEDCMZHSIOSIYRJF4X2AZQP6/events.json","paper":"https://pith.science/paper/TBGEDCMZ"},"agent_actions":{"view_html":"https://pith.science/pith/TBGEDCMZHSIOSIYRJF4X2AZQP6","download_json":"https://pith.science/pith/TBGEDCMZHSIOSIYRJF4X2AZQP6.json","view_paper":"https://pith.science/paper/TBGEDCMZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.11132&json=true","fetch_graph":"https://pith.science/api/pith-number/TBGEDCMZHSIOSIYRJF4X2AZQP6/graph.json","fetch_events":"https://pith.science/api/pith-number/TBGEDCMZHSIOSIYRJF4X2AZQP6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TBGEDCMZHSIOSIYRJF4X2AZQP6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TBGEDCMZHSIOSIYRJF4X2AZQP6/action/storage_attestation","attest_author":"https://pith.science/pith/TBGEDCMZHSIOSIYRJF4X2AZQP6/action/author_attestation","sign_citation":"https://pith.science/pith/TBGEDCMZHSIOSIYRJF4X2AZQP6/action/citation_signature","submit_replication":"https://pith.science/pith/TBGEDCMZHSIOSIYRJF4X2AZQP6/action/replication_record"}},"created_at":"2026-08-12T01:24:39.926025+00:00","updated_at":"2026-08-12T01:24:39.926025+00:00"}