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For an integer $k\\ge 1$, let ${\\mathbb G}=\\{G_k,G_{k+1},\\dots \\}$ be a class of graphs with $\\delta(G_n)\\ge 1$. We define the critical Ramsey number $R_{\\mathbb G}(F,H)$ with respect to $\\mathbb G$ to be $\\max\\{n|\\; K_r\\setminus G_n \\rightarrow(F,H),\\,G_n\\in{\\mathbb G}\\big\\}$, where $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":"1712.03226","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-08T03:34:12Z","cross_cats_sorted":[],"title_canon_sha256":"152ff42199f0da95d6d74f40a5f836790f7dea8fd557519dae33e973d36aba12","abstract_canon_sha256":"eb6ca18fe0679c6800d8dafc88fa4dffc8d6c5affedb393407085fe3fdf1954d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:22.824402Z","signature_b64":"z35hfzNZrnIWWUHzybGqcr4WbkiUkkdjVcz/9xuDhjfGZWqH1m+1w0iXeG+BI2H4v3tifBFOx4TRJTXDoJtaCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"611bc99e3f071e625991ddde32e469659b1699a9d13f8ddc4e756d1166aeddf3","last_reissued_at":"2026-05-17T23:48:22.823947Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:22.823947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Redundant edges in Ramsey graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yan Li, Ye Wang, Yusheng Li","submitted_at":"2017-12-08T03:34:12Z","abstract_excerpt":"For graphs $G$, $F$ and $H$, let $G\\rightarrow (F,H)$ signify that any edge coloring of $G$ in red and blue contains a red $F$ or a blue $H$. The Ramsey number $R(F,H)=\\min\\{r|\\; K_r\\rightarrow (F,H)\\}$. In this note, we consider redundant edges in Ramsey graphs, which are associate with critical Ramsey numbers. For an integer $k\\ge 1$, let ${\\mathbb G}=\\{G_k,G_{k+1},\\dots \\}$ be a class of graphs with $\\delta(G_n)\\ge 1$. We define the critical Ramsey number $R_{\\mathbb G}(F,H)$ with respect to $\\mathbb G$ to be $\\max\\{n|\\; K_r\\setminus G_n \\rightarrow(F,H),\\,G_n\\in{\\mathbb G}\\big\\}$, where $r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.03226","kind":"arxiv","version":5},"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":"1712.03226","created_at":"2026-05-17T23:48:22.824024+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.03226v5","created_at":"2026-05-17T23:48:22.824024+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.03226","created_at":"2026-05-17T23:48:22.824024+00:00"},{"alias_kind":"pith_short_12","alias_value":"MEN4THR7A4PG","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_16","alias_value":"MEN4THR7A4PGEWMR","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_8","alias_value":"MEN4THR7","created_at":"2026-05-18T12:31:31.346846+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/MEN4THR7A4PGEWMR3XPDFZDJMW","json":"https://pith.science/pith/MEN4THR7A4PGEWMR3XPDFZDJMW.json","graph_json":"https://pith.science/api/pith-number/MEN4THR7A4PGEWMR3XPDFZDJMW/graph.json","events_json":"https://pith.science/api/pith-number/MEN4THR7A4PGEWMR3XPDFZDJMW/events.json","paper":"https://pith.science/paper/MEN4THR7"},"agent_actions":{"view_html":"https://pith.science/pith/MEN4THR7A4PGEWMR3XPDFZDJMW","download_json":"https://pith.science/pith/MEN4THR7A4PGEWMR3XPDFZDJMW.json","view_paper":"https://pith.science/paper/MEN4THR7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.03226&json=true","fetch_graph":"https://pith.science/api/pith-number/MEN4THR7A4PGEWMR3XPDFZDJMW/graph.json","fetch_events":"https://pith.science/api/pith-number/MEN4THR7A4PGEWMR3XPDFZDJMW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MEN4THR7A4PGEWMR3XPDFZDJMW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MEN4THR7A4PGEWMR3XPDFZDJMW/action/storage_attestation","attest_author":"https://pith.science/pith/MEN4THR7A4PGEWMR3XPDFZDJMW/action/author_attestation","sign_citation":"https://pith.science/pith/MEN4THR7A4PGEWMR3XPDFZDJMW/action/citation_signature","submit_replication":"https://pith.science/pith/MEN4THR7A4PGEWMR3XPDFZDJMW/action/replication_record"}},"created_at":"2026-05-17T23:48:22.824024+00:00","updated_at":"2026-05-17T23:48:22.824024+00:00"}