{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:WWEWGAQOJTAVFNTOS4SW6NJKR5","short_pith_number":"pith:WWEWGAQO","schema_version":"1.0","canonical_sha256":"b58963020e4cc152b66e97256f352a8f53f7cfeaa97c27a954fe58fe7c31beb6","source":{"kind":"arxiv","id":"1905.08123","version":1},"attestation_state":"computed","paper":{"title":"Sharp results concerning disjoint cross-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Andrey Kupavskii, Peter Frankl","submitted_at":"2019-05-20T14:02:08Z","abstract_excerpt":"For an $n$-element set $X$ let $\\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\\mathcal A$ and $\\mathcal B$ are called cross-intersecting if $A\\cap B \\neq \\emptyset$ holds for all $A\\in\\mathcal A$, $B\\in\\mathcal B$. Let $f(n,k)$ denote the maximum of $\\min\\{|\\mathcal A|, |\\mathcal B|\\}$ where the maximum is taken over all pairs of {\\em disjoint}, cross-intersecting families $\\mathcal A, \\mathcal B\\subset\\binom{[n]}{k}$. Let $c=\\log_2e$. We prove that $f(n,k)=\\left\\lfloor\\frac12\\binom{n-1}{k-1}\\right\\rfloor$ essentially iff $n>ck^2$ (cf. Theorem~1.4 for the exact "},"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":"1905.08123","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-20T14:02:08Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"e2b4f60499539c7003176a72b67f9f0dca6be882671693643c0e9ec20600ff82","abstract_canon_sha256":"f4d0ed6285a900f63cb506ac000287445afc89563f552a0e1b5f10f10198298d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:47.529751Z","signature_b64":"iYYC9syNe7Mxk+AFUmiN+6azMBIzHv8dCx/p0hAmyLiD3sVSePL42FBuyOIJYxGJl89pEWO5RF1HWlca1vpaBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b58963020e4cc152b66e97256f352a8f53f7cfeaa97c27a954fe58fe7c31beb6","last_reissued_at":"2026-05-17T23:45:47.529149Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:47.529149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp results concerning disjoint cross-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Andrey Kupavskii, Peter Frankl","submitted_at":"2019-05-20T14:02:08Z","abstract_excerpt":"For an $n$-element set $X$ let $\\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\\mathcal A$ and $\\mathcal B$ are called cross-intersecting if $A\\cap B \\neq \\emptyset$ holds for all $A\\in\\mathcal A$, $B\\in\\mathcal B$. Let $f(n,k)$ denote the maximum of $\\min\\{|\\mathcal A|, |\\mathcal B|\\}$ where the maximum is taken over all pairs of {\\em disjoint}, cross-intersecting families $\\mathcal A, \\mathcal B\\subset\\binom{[n]}{k}$. Let $c=\\log_2e$. We prove that $f(n,k)=\\left\\lfloor\\frac12\\binom{n-1}{k-1}\\right\\rfloor$ essentially iff $n>ck^2$ (cf. Theorem~1.4 for the exact "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.08123","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":"1905.08123","created_at":"2026-05-17T23:45:47.529226+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.08123v1","created_at":"2026-05-17T23:45:47.529226+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.08123","created_at":"2026-05-17T23:45:47.529226+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/WWEWGAQOJTAVFNTOS4SW6NJKR5","json":"https://pith.science/pith/WWEWGAQOJTAVFNTOS4SW6NJKR5.json","graph_json":"https://pith.science/api/pith-number/WWEWGAQOJTAVFNTOS4SW6NJKR5/graph.json","events_json":"https://pith.science/api/pith-number/WWEWGAQOJTAVFNTOS4SW6NJKR5/events.json","paper":"https://pith.science/paper/WWEWGAQO"},"agent_actions":{"view_html":"https://pith.science/pith/WWEWGAQOJTAVFNTOS4SW6NJKR5","download_json":"https://pith.science/pith/WWEWGAQOJTAVFNTOS4SW6NJKR5.json","view_paper":"https://pith.science/paper/WWEWGAQO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.08123&json=true","fetch_graph":"https://pith.science/api/pith-number/WWEWGAQOJTAVFNTOS4SW6NJKR5/graph.json","fetch_events":"https://pith.science/api/pith-number/WWEWGAQOJTAVFNTOS4SW6NJKR5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WWEWGAQOJTAVFNTOS4SW6NJKR5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WWEWGAQOJTAVFNTOS4SW6NJKR5/action/storage_attestation","attest_author":"https://pith.science/pith/WWEWGAQOJTAVFNTOS4SW6NJKR5/action/author_attestation","sign_citation":"https://pith.science/pith/WWEWGAQOJTAVFNTOS4SW6NJKR5/action/citation_signature","submit_replication":"https://pith.science/pith/WWEWGAQOJTAVFNTOS4SW6NJKR5/action/replication_record"}},"created_at":"2026-05-17T23:45:47.529226+00:00","updated_at":"2026-05-17T23:45:47.529226+00:00"}