{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:WWEWGAQOJTAVFNTOS4SW6NJKR5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f4d0ed6285a900f63cb506ac000287445afc89563f552a0e1b5f10f10198298d","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-20T14:02:08Z","title_canon_sha256":"e2b4f60499539c7003176a72b67f9f0dca6be882671693643c0e9ec20600ff82"},"schema_version":"1.0","source":{"id":"1905.08123","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.08123","created_at":"2026-05-17T23:45:47Z"},{"alias_kind":"arxiv_version","alias_value":"1905.08123v1","created_at":"2026-05-17T23:45:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.08123","created_at":"2026-05-17T23:45:47Z"},{"alias_kind":"pith_short_12","alias_value":"WWEWGAQOJTAV","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"WWEWGAQOJTAVFNTO","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"WWEWGAQO","created_at":"2026-05-18T12:33:33Z"}],"graph_snapshots":[{"event_id":"sha256:7499e214b19973ea7ec984d5b17fc9250ada3aadf13a1ef2f1b3046f207fc5b8","target":"graph","created_at":"2026-05-17T23:45:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 ","authors_text":"Andrey Kupavskii, Peter Frankl","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-20T14:02:08Z","title":"Sharp results concerning disjoint cross-intersecting families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.08123","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:351944ac37ef6c714224242787b37f234a40e3fe452cc9e5a0b557e33f0fe150","target":"record","created_at":"2026-05-17T23:45:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f4d0ed6285a900f63cb506ac000287445afc89563f552a0e1b5f10f10198298d","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-20T14:02:08Z","title_canon_sha256":"e2b4f60499539c7003176a72b67f9f0dca6be882671693643c0e9ec20600ff82"},"schema_version":"1.0","source":{"id":"1905.08123","kind":"arxiv","version":1}},"canonical_sha256":"b58963020e4cc152b66e97256f352a8f53f7cfeaa97c27a954fe58fe7c31beb6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b58963020e4cc152b66e97256f352a8f53f7cfeaa97c27a954fe58fe7c31beb6","first_computed_at":"2026-05-17T23:45:47.529149Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:45:47.529149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iYYC9syNe7Mxk+AFUmiN+6azMBIzHv8dCx/p0hAmyLiD3sVSePL42FBuyOIJYxGJl89pEWO5RF1HWlca1vpaBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:45:47.529751Z","signed_message":"canonical_sha256_bytes"},"source_id":"1905.08123","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:351944ac37ef6c714224242787b37f234a40e3fe452cc9e5a0b557e33f0fe150","sha256:7499e214b19973ea7ec984d5b17fc9250ada3aadf13a1ef2f1b3046f207fc5b8"],"state_sha256":"88506de3b511cc21d9c9003de3bc998d59168e3ad4f965bb1e4726870b5f26d8"}