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Then $G$ is said to be $H$-exact if any induced subgraph of $G$ on $k$ vertices is either isomorphic to $H$ or incomparable with $H$. Exact($H$) is the family of all graphs $G$ which are $H$-exact.\n  We pose the following problem: For a graph $H$ on $k$ vertices, determine or estimate $f(H) = \\max \\{n: \\exists G \\in \\text{Ex"},"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":"2605.22077","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-21T07:16:34Z","cross_cats_sorted":[],"title_canon_sha256":"ac739a7ec96572be1404bdd7cb2f342f560e134aee9b22a967c8cd88ea145492","abstract_canon_sha256":"51445bf9c1d7dec94a555dab235d38c13073579e93d6737a660555deb96d4ef4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:24.049365Z","signature_b64":"+jNXNjc0qxJIquSpUPspsAvvF+kkPuG9zh0r1MvyQyryxvsaoH8DDY+j4wHJCFpsSMHM9hgRy5iq99eL8qiJAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"147428f383d4833f4ab0175de29520ab65a855f95a3845cb711df8fffee9d484","last_reissued_at":"2026-05-22T01:04:24.048390Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:24.048390Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Induced/Incomparable versus Ramsey","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christina Zarb, Yair Caro, Zsolt Tuza","submitted_at":"2026-05-21T07:16:34Z","abstract_excerpt":"We consider the following problem: Let $H$ and $F$ be two graphs on $k$ vertices and assume $F \\neq H$. 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Exact($H$) is the family of all graphs $G$ which are $H$-exact.\n  We pose the following problem: For a graph $H$ on $k$ vertices, determine or estimate $f(H) = \\max \\{n: \\exists G \\in \\text{Ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22077","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/2605.22077/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":"2605.22077","created_at":"2026-05-22T01:04:24.048537+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.22077v1","created_at":"2026-05-22T01:04:24.048537+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.22077","created_at":"2026-05-22T01:04:24.048537+00:00"},{"alias_kind":"pith_short_12","alias_value":"CR2CR44D2SBT","created_at":"2026-05-22T01:04:24.048537+00:00"},{"alias_kind":"pith_short_16","alias_value":"CR2CR44D2SBT6SVQ","created_at":"2026-05-22T01:04:24.048537+00:00"},{"alias_kind":"pith_short_8","alias_value":"CR2CR44D","created_at":"2026-05-22T01:04:24.048537+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/CR2CR44D2SBT6SVQC5O6FFJAVN","json":"https://pith.science/pith/CR2CR44D2SBT6SVQC5O6FFJAVN.json","graph_json":"https://pith.science/api/pith-number/CR2CR44D2SBT6SVQC5O6FFJAVN/graph.json","events_json":"https://pith.science/api/pith-number/CR2CR44D2SBT6SVQC5O6FFJAVN/events.json","paper":"https://pith.science/paper/CR2CR44D"},"agent_actions":{"view_html":"https://pith.science/pith/CR2CR44D2SBT6SVQC5O6FFJAVN","download_json":"https://pith.science/pith/CR2CR44D2SBT6SVQC5O6FFJAVN.json","view_paper":"https://pith.science/paper/CR2CR44D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.22077&json=true","fetch_graph":"https://pith.science/api/pith-number/CR2CR44D2SBT6SVQC5O6FFJAVN/graph.json","fetch_events":"https://pith.science/api/pith-number/CR2CR44D2SBT6SVQC5O6FFJAVN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CR2CR44D2SBT6SVQC5O6FFJAVN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CR2CR44D2SBT6SVQC5O6FFJAVN/action/storage_attestation","attest_author":"https://pith.science/pith/CR2CR44D2SBT6SVQC5O6FFJAVN/action/author_attestation","sign_citation":"https://pith.science/pith/CR2CR44D2SBT6SVQC5O6FFJAVN/action/citation_signature","submit_replication":"https://pith.science/pith/CR2CR44D2SBT6SVQC5O6FFJAVN/action/replication_record"}},"created_at":"2026-05-22T01:04:24.048537+00:00","updated_at":"2026-05-22T01:04:24.048537+00:00"}