{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:42C6UZQARXTCRVLY3M4BQ6OPKP","short_pith_number":"pith:42C6UZQA","schema_version":"1.0","canonical_sha256":"e685ea66008de628d578db381879cf53ff626bd492df794b6c2c5ff4f42235da","source":{"kind":"arxiv","id":"2507.09832","version":1},"attestation_state":"computed","paper":{"title":"Fan-goodness of sparse graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ting Huang, Yanbo Zhang, Yaojun Chen","submitted_at":"2025-07-13T23:59:39Z","abstract_excerpt":"Let $G$ be a connected graph of order $n$, $F_k$ be a fan consisting of $k$ triangles sharing a common vertex, and $tF_k$ be $t$ vertex-disjoint copies of $F_k$. Brennan (2017) showed the Ramsey number $r(G,F_k)=2n-1$ for $G$ being a unicyclic graph for $n \\geq k^2-k+1$ and $k\\ge 18$, and asked the threshold $c(n)$ for which $r(G,F_k) \\geq 2n$ holds for any $G$ containing at least $c(n)$ cycles and $n$ being large. In this paper, we consider fan-goodness of general sparse graphs and show that if $G$ has at most $n(1+\\epsilon(k))$ edges, where $\\epsilon(k)$ is a constant depending on $k$, then "},"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":"2507.09832","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-07-13T23:59:39Z","cross_cats_sorted":[],"title_canon_sha256":"650078dd8663bff5e592693bc249170099ad6afcf29d82f8cc47ef09deeb451e","abstract_canon_sha256":"c42e03f7109dc7d06075a67493264336550d7bbc8555757c2a11486b235b7222"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:36:40.487795Z","signature_b64":"7/7jbQV5QSk/UCmnHFfteWl88u78ZgUp8QBcKt94K6GIBuEC8c4jeuGLXT49Ss/EB2FM0fvTjWIJQE6GId8iCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e685ea66008de628d578db381879cf53ff626bd492df794b6c2c5ff4f42235da","last_reissued_at":"2026-07-05T11:36:40.487326Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:36:40.487326Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fan-goodness of sparse graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ting Huang, Yanbo Zhang, Yaojun Chen","submitted_at":"2025-07-13T23:59:39Z","abstract_excerpt":"Let $G$ be a connected graph of order $n$, $F_k$ be a fan consisting of $k$ triangles sharing a common vertex, and $tF_k$ be $t$ vertex-disjoint copies of $F_k$. Brennan (2017) showed the Ramsey number $r(G,F_k)=2n-1$ for $G$ being a unicyclic graph for $n \\geq k^2-k+1$ and $k\\ge 18$, and asked the threshold $c(n)$ for which $r(G,F_k) \\geq 2n$ holds for any $G$ containing at least $c(n)$ cycles and $n$ being large. In this paper, we consider fan-goodness of general sparse graphs and show that if $G$ has at most $n(1+\\epsilon(k))$ edges, where $\\epsilon(k)$ is a constant depending on $k$, then "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.09832","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/2507.09832/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":"2507.09832","created_at":"2026-07-05T11:36:40.487389+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.09832v1","created_at":"2026-07-05T11:36:40.487389+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.09832","created_at":"2026-07-05T11:36:40.487389+00:00"},{"alias_kind":"pith_short_12","alias_value":"42C6UZQARXTC","created_at":"2026-07-05T11:36:40.487389+00:00"},{"alias_kind":"pith_short_16","alias_value":"42C6UZQARXTCRVLY","created_at":"2026-07-05T11:36:40.487389+00:00"},{"alias_kind":"pith_short_8","alias_value":"42C6UZQA","created_at":"2026-07-05T11:36:40.487389+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.17602","citing_title":"AutoRubric-T2I: Robust Rule-Based Reward Model for Text-to-Image Alignment","ref_index":22,"is_internal_anchor":false},{"citing_arxiv_id":"2605.17602","citing_title":"AutoRubric-T2I: Robust Rule-Based Reward Model for Text-to-Image Alignment","ref_index":22,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/42C6UZQARXTCRVLY3M4BQ6OPKP","json":"https://pith.science/pith/42C6UZQARXTCRVLY3M4BQ6OPKP.json","graph_json":"https://pith.science/api/pith-number/42C6UZQARXTCRVLY3M4BQ6OPKP/graph.json","events_json":"https://pith.science/api/pith-number/42C6UZQARXTCRVLY3M4BQ6OPKP/events.json","paper":"https://pith.science/paper/42C6UZQA"},"agent_actions":{"view_html":"https://pith.science/pith/42C6UZQARXTCRVLY3M4BQ6OPKP","download_json":"https://pith.science/pith/42C6UZQARXTCRVLY3M4BQ6OPKP.json","view_paper":"https://pith.science/paper/42C6UZQA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.09832&json=true","fetch_graph":"https://pith.science/api/pith-number/42C6UZQARXTCRVLY3M4BQ6OPKP/graph.json","fetch_events":"https://pith.science/api/pith-number/42C6UZQARXTCRVLY3M4BQ6OPKP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/42C6UZQARXTCRVLY3M4BQ6OPKP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/42C6UZQARXTCRVLY3M4BQ6OPKP/action/storage_attestation","attest_author":"https://pith.science/pith/42C6UZQARXTCRVLY3M4BQ6OPKP/action/author_attestation","sign_citation":"https://pith.science/pith/42C6UZQARXTCRVLY3M4BQ6OPKP/action/citation_signature","submit_replication":"https://pith.science/pith/42C6UZQARXTCRVLY3M4BQ6OPKP/action/replication_record"}},"created_at":"2026-07-05T11:36:40.487389+00:00","updated_at":"2026-07-05T11:36:40.487389+00:00"}