{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:7SIPZ4B7OTGG26ZATPYNYZAOVG","short_pith_number":"pith:7SIPZ4B7","schema_version":"1.0","canonical_sha256":"fc90fcf03f74cc6d7b209bf0dc640ea99d9122bfe9712d91fa1c4271efb7fde2","source":{"kind":"arxiv","id":"2407.03121","version":1},"attestation_state":"computed","paper":{"title":"Erd\\H{o}s-Rogers functions for arbitrary pairs of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dhruv Mubayi, Jacques Verstraete","submitted_at":"2024-07-03T14:06:04Z","abstract_excerpt":"Let $f_{F,G}(n)$ be the largest size of an induced $F$-free subgraph that every $n$-vertex $G$-free graph is guaranteed to contain. We prove that for any triangle-free graph $F$, \\[ f_{F,K_3}(n) = f_{K_2,K_3}(n)^{1 + o(1)} = n^{\\frac{1}{2} + o(1)}.\\] Along the way we give a slight improvement of a construction of Erd\\H os-Frankl-R\\\"odl for the Brown-Erd\\H os-S\\'os $(3r-3,3)$-problem when $r$ is large.\n  In contrast to our result for $K_3$, for any $K_4$-free graph $F$ containing a cycle, we prove there exists $c_F > 0$ such that $$f_{F,K_4}(n) > f_{K_2,K_4}(n)^{1 + c_F} = n^{\\frac{1}{3}+c_F+o("},"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":"2407.03121","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-07-03T14:06:04Z","cross_cats_sorted":[],"title_canon_sha256":"fa7583d4ee55b2e98b9dbd4f12ca06afdba2a167e18dad0cbcce27f89256a8b2","abstract_canon_sha256":"dfacf8e70ebb7058941ea9baedfe8ba3a3719ed808dcc597ab517b7abf1881b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:39:46.639661Z","signature_b64":"P4RTzpp0CVyftmeTsEfeNqlQulXMmjaBg9wOZ932HKBagXoME2X7VW0kp8z9EyeqU+d1HqhzN8q/a/qnWDPLDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc90fcf03f74cc6d7b209bf0dc640ea99d9122bfe9712d91fa1c4271efb7fde2","last_reissued_at":"2026-07-05T08:39:46.639188Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:39:46.639188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Erd\\H{o}s-Rogers functions for arbitrary pairs of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dhruv Mubayi, Jacques Verstraete","submitted_at":"2024-07-03T14:06:04Z","abstract_excerpt":"Let $f_{F,G}(n)$ be the largest size of an induced $F$-free subgraph that every $n$-vertex $G$-free graph is guaranteed to contain. We prove that for any triangle-free graph $F$, \\[ f_{F,K_3}(n) = f_{K_2,K_3}(n)^{1 + o(1)} = n^{\\frac{1}{2} + o(1)}.\\] Along the way we give a slight improvement of a construction of Erd\\H os-Frankl-R\\\"odl for the Brown-Erd\\H os-S\\'os $(3r-3,3)$-problem when $r$ is large.\n  In contrast to our result for $K_3$, for any $K_4$-free graph $F$ containing a cycle, we prove there exists $c_F > 0$ such that $$f_{F,K_4}(n) > f_{K_2,K_4}(n)^{1 + c_F} = n^{\\frac{1}{3}+c_F+o("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.03121","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/2407.03121/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":"2407.03121","created_at":"2026-07-05T08:39:46.639244+00:00"},{"alias_kind":"arxiv_version","alias_value":"2407.03121v1","created_at":"2026-07-05T08:39:46.639244+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.03121","created_at":"2026-07-05T08:39:46.639244+00:00"},{"alias_kind":"pith_short_12","alias_value":"7SIPZ4B7OTGG","created_at":"2026-07-05T08:39:46.639244+00:00"},{"alias_kind":"pith_short_16","alias_value":"7SIPZ4B7OTGG26ZA","created_at":"2026-07-05T08:39:46.639244+00:00"},{"alias_kind":"pith_short_8","alias_value":"7SIPZ4B7","created_at":"2026-07-05T08:39:46.639244+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.02835","citing_title":"A Note on Generalized Erd\\H{o}s-Rogers Problems","ref_index":44,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7SIPZ4B7OTGG26ZATPYNYZAOVG","json":"https://pith.science/pith/7SIPZ4B7OTGG26ZATPYNYZAOVG.json","graph_json":"https://pith.science/api/pith-number/7SIPZ4B7OTGG26ZATPYNYZAOVG/graph.json","events_json":"https://pith.science/api/pith-number/7SIPZ4B7OTGG26ZATPYNYZAOVG/events.json","paper":"https://pith.science/paper/7SIPZ4B7"},"agent_actions":{"view_html":"https://pith.science/pith/7SIPZ4B7OTGG26ZATPYNYZAOVG","download_json":"https://pith.science/pith/7SIPZ4B7OTGG26ZATPYNYZAOVG.json","view_paper":"https://pith.science/paper/7SIPZ4B7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2407.03121&json=true","fetch_graph":"https://pith.science/api/pith-number/7SIPZ4B7OTGG26ZATPYNYZAOVG/graph.json","fetch_events":"https://pith.science/api/pith-number/7SIPZ4B7OTGG26ZATPYNYZAOVG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7SIPZ4B7OTGG26ZATPYNYZAOVG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7SIPZ4B7OTGG26ZATPYNYZAOVG/action/storage_attestation","attest_author":"https://pith.science/pith/7SIPZ4B7OTGG26ZATPYNYZAOVG/action/author_attestation","sign_citation":"https://pith.science/pith/7SIPZ4B7OTGG26ZATPYNYZAOVG/action/citation_signature","submit_replication":"https://pith.science/pith/7SIPZ4B7OTGG26ZATPYNYZAOVG/action/replication_record"}},"created_at":"2026-07-05T08:39:46.639244+00:00","updated_at":"2026-07-05T08:39:46.639244+00:00"}