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We prove that for any fixed $k$ if $n$ is large enough, then $F(n,k),f(n,k)=2^{(1/2+o(1))n}$ holds.\n  We also introduce the general functions for any poset $P$ and integer $c\\ge |P|$: let $F(n,c,P)$ ($f(n,c,P)$) denote the the maximum possible size of the smallest color class in a (partial) $c$-coloring of the Boolean lattice $B_n"},"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":"1812.09058","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-21T11:27:48Z","cross_cats_sorted":[],"title_canon_sha256":"d833f4679a9401cb48f8ffe346eee0a3f813e3cf8d21e972747fe203bde25f99","abstract_canon_sha256":"a565b9c0ccc15fabc3424c8a3cbb0768feb5064f8e92b6241c84e0c1b744760f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:45.532057Z","signature_b64":"O4P1gqewpFZZg2JBM5C69WvoOXhaz6MqR+mQlppQ7n26fW/lHsLbQHx8Z9y3jJnD0gXt/onsv3Nf7SAmi7cFDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9c790cc4826d7a63ae90459bd1be8a7874a06ce39a26aab370c85c9c168e5b6","last_reissued_at":"2026-05-17T23:57:45.531553Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:45.531553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On colorings of the Boolean lattice avoiding a rainbow copy of a poset","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Patk\\'os","submitted_at":"2018-12-21T11:27:48Z","abstract_excerpt":"Let $F(n,k)$ ($f(n,k)$) denote the maximum possible size of the smallest color class in a (partial) $k$-coloring of the Boolean lattice $B_n$ that does not admit a rainbow antichain of size $k$. 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We prove that for any fixed $k$ if $n$ is large enough, then $F(n,k),f(n,k)=2^{(1/2+o(1))n}$ holds.\n  We also introduce the general functions for any poset $P$ and integer $c\\ge |P|$: let $F(n,c,P)$ ($f(n,c,P)$) denote the the maximum possible size of the smallest color class in a (partial) $c$-coloring of the Boolean lattice $B_n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.09058","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":""},"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":"1812.09058","created_at":"2026-05-17T23:57:45.531649+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.09058v1","created_at":"2026-05-17T23:57:45.531649+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.09058","created_at":"2026-05-17T23:57:45.531649+00:00"},{"alias_kind":"pith_short_12","alias_value":"XHDZBTCIE3L2","created_at":"2026-05-18T12:33:01.666342+00:00"},{"alias_kind":"pith_short_16","alias_value":"XHDZBTCIE3L2MOXJ","created_at":"2026-05-18T12:33:01.666342+00:00"},{"alias_kind":"pith_short_8","alias_value":"XHDZBTCI","created_at":"2026-05-18T12:33:01.666342+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/XHDZBTCIE3L2MOXJARM32G7IU6","json":"https://pith.science/pith/XHDZBTCIE3L2MOXJARM32G7IU6.json","graph_json":"https://pith.science/api/pith-number/XHDZBTCIE3L2MOXJARM32G7IU6/graph.json","events_json":"https://pith.science/api/pith-number/XHDZBTCIE3L2MOXJARM32G7IU6/events.json","paper":"https://pith.science/paper/XHDZBTCI"},"agent_actions":{"view_html":"https://pith.science/pith/XHDZBTCIE3L2MOXJARM32G7IU6","download_json":"https://pith.science/pith/XHDZBTCIE3L2MOXJARM32G7IU6.json","view_paper":"https://pith.science/paper/XHDZBTCI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.09058&json=true","fetch_graph":"https://pith.science/api/pith-number/XHDZBTCIE3L2MOXJARM32G7IU6/graph.json","fetch_events":"https://pith.science/api/pith-number/XHDZBTCIE3L2MOXJARM32G7IU6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XHDZBTCIE3L2MOXJARM32G7IU6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XHDZBTCIE3L2MOXJARM32G7IU6/action/storage_attestation","attest_author":"https://pith.science/pith/XHDZBTCIE3L2MOXJARM32G7IU6/action/author_attestation","sign_citation":"https://pith.science/pith/XHDZBTCIE3L2MOXJARM32G7IU6/action/citation_signature","submit_replication":"https://pith.science/pith/XHDZBTCIE3L2MOXJARM32G7IU6/action/replication_record"}},"created_at":"2026-05-17T23:57:45.531649+00:00","updated_at":"2026-05-17T23:57:45.531649+00:00"}