{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:5HJQZL5BJGSHPUBA5EO5XBKH2Y","short_pith_number":"pith:5HJQZL5B","schema_version":"1.0","canonical_sha256":"e9d30cafa149a477d020e91ddb8547d612f76e6a7f453b404e59bcf57daa17c8","source":{"kind":"arxiv","id":"2607.00456","version":1},"attestation_state":"computed","paper":{"title":"Multiplicity for partially ordered sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gyula O.H. Katona, Yaping Mao","submitted_at":"2026-07-01T05:23:17Z","abstract_excerpt":"Let $\\mathcal Q=\\{Q_a:a\\geq1\\}$ be a nested family of finite posets such that $Q_a\\subseteq Q_{a+1}$ and $|Q_a|<|Q_{a+1}|$. For a poset $Q$, let $\\mathcal C_t(Q)$ denote the set of all strict $t$-chains in $Q$. Given an $r$-coloring of $\\mathcal C_t(Q_a)$ and posets $P_1,\\ldots,P_r$, a weak copy of $P_i$ is called monochromatic of color $i$ if all $t$-chains in the copy have color $i$; the strong version is defined in the same way for induced copies. The corresponding weak and strong multiplicity parameters are the minimum possible total number of such monochromatic copies in the host poset.Fo"},"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":"2607.00456","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-01T05:23:17Z","cross_cats_sorted":[],"title_canon_sha256":"67ec655e2766592a39394a450f14161437189abf25ded43f9dc54acf867e6857","abstract_canon_sha256":"84ed7bd74474a53aa7e8df246d67cc18c5b685043e6b63b085acaafc8f17f78f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-02T01:17:43.956920Z","signature_b64":"mfH90TvmvGPQ7dgG672Gwea3fXfgsgNXYjyr0/g7Za087dTyVJHnROQxy2WB1aBAWnoiSDMh2oyWsXAsk5EtDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e9d30cafa149a477d020e91ddb8547d612f76e6a7f453b404e59bcf57daa17c8","last_reissued_at":"2026-07-02T01:17:43.956529Z","signature_status":"signed_v1","first_computed_at":"2026-07-02T01:17:43.956529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiplicity for partially ordered sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gyula O.H. Katona, Yaping Mao","submitted_at":"2026-07-01T05:23:17Z","abstract_excerpt":"Let $\\mathcal Q=\\{Q_a:a\\geq1\\}$ be a nested family of finite posets such that $Q_a\\subseteq Q_{a+1}$ and $|Q_a|<|Q_{a+1}|$. For a poset $Q$, let $\\mathcal C_t(Q)$ denote the set of all strict $t$-chains in $Q$. Given an $r$-coloring of $\\mathcal C_t(Q_a)$ and posets $P_1,\\ldots,P_r$, a weak copy of $P_i$ is called monochromatic of color $i$ if all $t$-chains in the copy have color $i$; the strong version is defined in the same way for induced copies. The corresponding weak and strong multiplicity parameters are the minimum possible total number of such monochromatic copies in the host poset.Fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00456","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/2607.00456/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":"2607.00456","created_at":"2026-07-02T01:17:43.956589+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.00456v1","created_at":"2026-07-02T01:17:43.956589+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.00456","created_at":"2026-07-02T01:17:43.956589+00:00"},{"alias_kind":"pith_short_12","alias_value":"5HJQZL5BJGSH","created_at":"2026-07-02T01:17:43.956589+00:00"},{"alias_kind":"pith_short_16","alias_value":"5HJQZL5BJGSHPUBA","created_at":"2026-07-02T01:17:43.956589+00:00"},{"alias_kind":"pith_short_8","alias_value":"5HJQZL5B","created_at":"2026-07-02T01:17:43.956589+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/5HJQZL5BJGSHPUBA5EO5XBKH2Y","json":"https://pith.science/pith/5HJQZL5BJGSHPUBA5EO5XBKH2Y.json","graph_json":"https://pith.science/api/pith-number/5HJQZL5BJGSHPUBA5EO5XBKH2Y/graph.json","events_json":"https://pith.science/api/pith-number/5HJQZL5BJGSHPUBA5EO5XBKH2Y/events.json","paper":"https://pith.science/paper/5HJQZL5B"},"agent_actions":{"view_html":"https://pith.science/pith/5HJQZL5BJGSHPUBA5EO5XBKH2Y","download_json":"https://pith.science/pith/5HJQZL5BJGSHPUBA5EO5XBKH2Y.json","view_paper":"https://pith.science/paper/5HJQZL5B","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.00456&json=true","fetch_graph":"https://pith.science/api/pith-number/5HJQZL5BJGSHPUBA5EO5XBKH2Y/graph.json","fetch_events":"https://pith.science/api/pith-number/5HJQZL5BJGSHPUBA5EO5XBKH2Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5HJQZL5BJGSHPUBA5EO5XBKH2Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5HJQZL5BJGSHPUBA5EO5XBKH2Y/action/storage_attestation","attest_author":"https://pith.science/pith/5HJQZL5BJGSHPUBA5EO5XBKH2Y/action/author_attestation","sign_citation":"https://pith.science/pith/5HJQZL5BJGSHPUBA5EO5XBKH2Y/action/citation_signature","submit_replication":"https://pith.science/pith/5HJQZL5BJGSHPUBA5EO5XBKH2Y/action/replication_record"}},"created_at":"2026-07-02T01:17:43.956589+00:00","updated_at":"2026-07-02T01:17:43.956589+00:00"}