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Let $K_{p+1}(\\alpha;n)$ denote the complete $(p+1)$-partite graph with $p$ partite sets of size $\\alpha$ and one partite set of size $n$. We determine all graphs $G$ for which $K_{p+1}(\\alpha;n)$ is $G$-good for large $n$. The characterization depends on the parameter $\\mathrm{snd}(\\alpha)$, the smallest non-divisor of $\\alpha$."},"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":"2605.26826","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-26T10:44:17Z","cross_cats_sorted":[],"title_canon_sha256":"4b0dbf8d9b1928823e91ccd177fb569bb489f74013bde3285c1b02626d766e8c","abstract_canon_sha256":"131f7380b223f049db644b75ae18777231bf91829cb46de7749d1935a41636b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-27T01:06:14.525910Z","signature_b64":"vZqBIZxUBo4pMFRDALUIsVFGn6Qbze+C1nEzfFtLO2ea0IUDoI79qek/pcjPNNxLGDbLX4k7XM/R8E1xkbciDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a96d4f9391c8ff8199a121dff775fd91ba098a98096c1d61a3dba8b134f9837","last_reissued_at":"2026-05-27T01:06:14.525093Z","signature_status":"signed_v1","first_computed_at":"2026-05-27T01:06:14.525093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ramsey goodness of complete multipartite graphs with one large part","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shaonan Mi, Ye Wang","submitted_at":"2026-05-26T10:44:17Z","abstract_excerpt":"For graph $G$, a connected graph $H$ of order $n$ is said to be $G$-good if $r(G,H)=(\\chi(G)-1)(n-1)+s(G)$, where $\\chi(G)$ is the chromatic number of $G$ and $s(G)$ is the minimum size of a color class in a $\\chi(G)$-coloring of $G$. Let $K_{p+1}(\\alpha;n)$ denote the complete $(p+1)$-partite graph with $p$ partite sets of size $\\alpha$ and one partite set of size $n$. We determine all graphs $G$ for which $K_{p+1}(\\alpha;n)$ is $G$-good for large $n$. The characterization depends on the parameter $\\mathrm{snd}(\\alpha)$, the smallest non-divisor of $\\alpha$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.26826","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/2605.26826/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":"2605.26826","created_at":"2026-05-27T01:06:14.525227+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.26826v1","created_at":"2026-05-27T01:06:14.525227+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.26826","created_at":"2026-05-27T01:06:14.525227+00:00"},{"alias_kind":"pith_short_12","alias_value":"LKLNJ6JZDSH7","created_at":"2026-05-27T01:06:14.525227+00:00"},{"alias_kind":"pith_short_16","alias_value":"LKLNJ6JZDSH7QGM2","created_at":"2026-05-27T01:06:14.525227+00:00"},{"alias_kind":"pith_short_8","alias_value":"LKLNJ6JZ","created_at":"2026-05-27T01:06:14.525227+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/LKLNJ6JZDSH7QGM2CIO765273E","json":"https://pith.science/pith/LKLNJ6JZDSH7QGM2CIO765273E.json","graph_json":"https://pith.science/api/pith-number/LKLNJ6JZDSH7QGM2CIO765273E/graph.json","events_json":"https://pith.science/api/pith-number/LKLNJ6JZDSH7QGM2CIO765273E/events.json","paper":"https://pith.science/paper/LKLNJ6JZ"},"agent_actions":{"view_html":"https://pith.science/pith/LKLNJ6JZDSH7QGM2CIO765273E","download_json":"https://pith.science/pith/LKLNJ6JZDSH7QGM2CIO765273E.json","view_paper":"https://pith.science/paper/LKLNJ6JZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.26826&json=true","fetch_graph":"https://pith.science/api/pith-number/LKLNJ6JZDSH7QGM2CIO765273E/graph.json","fetch_events":"https://pith.science/api/pith-number/LKLNJ6JZDSH7QGM2CIO765273E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LKLNJ6JZDSH7QGM2CIO765273E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LKLNJ6JZDSH7QGM2CIO765273E/action/storage_attestation","attest_author":"https://pith.science/pith/LKLNJ6JZDSH7QGM2CIO765273E/action/author_attestation","sign_citation":"https://pith.science/pith/LKLNJ6JZDSH7QGM2CIO765273E/action/citation_signature","submit_replication":"https://pith.science/pith/LKLNJ6JZDSH7QGM2CIO765273E/action/replication_record"}},"created_at":"2026-05-27T01:06:14.525227+00:00","updated_at":"2026-05-27T01:06:14.525227+00:00"}