{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:JX7ZT7SLC35UCRBPM4RCILWTRR","short_pith_number":"pith:JX7ZT7SL","schema_version":"1.0","canonical_sha256":"4dff99fe4b16fb41442f6722242ed38c761d6eb2be63d4168aeb25ece8b7cc43","source":{"kind":"arxiv","id":"1101.4417","version":2},"attestation_state":"computed","paper":{"title":"Critical graphs without triangles: an optimum density construction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wesley Pegden","submitted_at":"2011-01-24T00:29:54Z","abstract_excerpt":"We construct dense, triangle-free, chromatic-critical graphs of chromatic number $k$ for all $k\\geq 4$. For $k\\geq 6$ our constructions have $> (\\frac{1}{4} -\\varepsilon)n^2$ edges, which is asymptotically best possible by Tur\\'an's theorem. We also demonstrate (nonconstructively) the existence of dense $k$-critical graphs avoiding all odd cycles of length $\\leq \\ell$ for any $\\ell$ and any $k\\geq 4$, again with a best possible density of $>(\\frac{1}{4} -\\varepsilon)n^2$ edges for $k\\geq 6$. The families of graphs without triangles or of given odd-girth are thus rare examples where we know the"},"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":"1101.4417","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-24T00:29:54Z","cross_cats_sorted":[],"title_canon_sha256":"dc93592dc3e268f74cd764477dff021fe2c6eafdb1f0be5b0dbed0212d4e5668","abstract_canon_sha256":"800f86aef3a53fffb8abe22e1411affab93119a19276f629b17034e59e43e813"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:59.601765Z","signature_b64":"WrwvoTEdydM+D4xmsyPqe39W2e/aoYzHYDuxNxM8s42GGTKeavfXw0FV1SawLNvr2ZbJgNFKNfTZ78oFNBIcBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4dff99fe4b16fb41442f6722242ed38c761d6eb2be63d4168aeb25ece8b7cc43","last_reissued_at":"2026-05-18T03:00:59.601158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:59.601158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical graphs without triangles: an optimum density construction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wesley Pegden","submitted_at":"2011-01-24T00:29:54Z","abstract_excerpt":"We construct dense, triangle-free, chromatic-critical graphs of chromatic number $k$ for all $k\\geq 4$. For $k\\geq 6$ our constructions have $> (\\frac{1}{4} -\\varepsilon)n^2$ edges, which is asymptotically best possible by Tur\\'an's theorem. We also demonstrate (nonconstructively) the existence of dense $k$-critical graphs avoiding all odd cycles of length $\\leq \\ell$ for any $\\ell$ and any $k\\geq 4$, again with a best possible density of $>(\\frac{1}{4} -\\varepsilon)n^2$ edges for $k\\geq 6$. The families of graphs without triangles or of given odd-girth are thus rare examples where we know the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.4417","kind":"arxiv","version":2},"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":"1101.4417","created_at":"2026-05-18T03:00:59.601258+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.4417v2","created_at":"2026-05-18T03:00:59.601258+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.4417","created_at":"2026-05-18T03:00:59.601258+00:00"},{"alias_kind":"pith_short_12","alias_value":"JX7ZT7SLC35U","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_16","alias_value":"JX7ZT7SLC35UCRBP","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_8","alias_value":"JX7ZT7SL","created_at":"2026-05-18T12:26:32.869790+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/JX7ZT7SLC35UCRBPM4RCILWTRR","json":"https://pith.science/pith/JX7ZT7SLC35UCRBPM4RCILWTRR.json","graph_json":"https://pith.science/api/pith-number/JX7ZT7SLC35UCRBPM4RCILWTRR/graph.json","events_json":"https://pith.science/api/pith-number/JX7ZT7SLC35UCRBPM4RCILWTRR/events.json","paper":"https://pith.science/paper/JX7ZT7SL"},"agent_actions":{"view_html":"https://pith.science/pith/JX7ZT7SLC35UCRBPM4RCILWTRR","download_json":"https://pith.science/pith/JX7ZT7SLC35UCRBPM4RCILWTRR.json","view_paper":"https://pith.science/paper/JX7ZT7SL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.4417&json=true","fetch_graph":"https://pith.science/api/pith-number/JX7ZT7SLC35UCRBPM4RCILWTRR/graph.json","fetch_events":"https://pith.science/api/pith-number/JX7ZT7SLC35UCRBPM4RCILWTRR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JX7ZT7SLC35UCRBPM4RCILWTRR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JX7ZT7SLC35UCRBPM4RCILWTRR/action/storage_attestation","attest_author":"https://pith.science/pith/JX7ZT7SLC35UCRBPM4RCILWTRR/action/author_attestation","sign_citation":"https://pith.science/pith/JX7ZT7SLC35UCRBPM4RCILWTRR/action/citation_signature","submit_replication":"https://pith.science/pith/JX7ZT7SLC35UCRBPM4RCILWTRR/action/replication_record"}},"created_at":"2026-05-18T03:00:59.601258+00:00","updated_at":"2026-05-18T03:00:59.601258+00:00"}